Differential Equations in Applications Amelkin Wc

Sciencefor Evervono

B. 13. AMenLKUB

JJ: lI«pepepeHD:n8JIbHMe ypa BHeHIIHB npano meHHHX

HSARTeJI&CTBO «Hayxai MOCKBa

v V Amel'kin

Differential Equationsin Applications

MirPublishersMoscow

Translated from the Russian by Eugene Yankovsky

First published 1990

Revised from the 1987 Russian edition

Ha aHeAUUCXOM neune

Printed in the Union of Soviet Socialist Republics

ISBN 5-03-000521-8

© H3,IJ;aTeJILCTBO «HaYKa». rJIaBHaJl peAaKI.\BJI.pU3BKo-MaTeMaTUqeCKQH JIHTepaTypLI, 1987

© English translation, Eugene Yankovsky, 1990

Contents

PrefaceChapter 1. Construction of Differential Models and Their Solutions

1.1 Whose Coffee Was Hotter?1.2 Steady-State Heat Flow .. .1.3 An Incident in a National Park1.4 Liquid Flow Out of Vessels. The

Water Clock. . ..1.5 Effectiveness of Advertising1.6 Supply and Demand1.7 Chemical Reactions .1.8 Differential Models in Ecology1.9 A Problem from the Mathematical

Theory of Epidemics1.10 The Pursuit Curve1.11 Combat Models .1.12 Why Are Pendulum Clocks Inaccu-

rate? . . .....1.13 The Cycloidal Clock . . .1.14 The Brachistochroae Problem1.15 The Arithmetic Mean, the Geo-

metric Mean, and the AssociatedDifferential Equation ..•.

1.16 On the Flight of an Object Thrownat an Angle to the Horizon

1.17 Weightlessness .•.•.....1.18 Kepler's Laws of Planetary Motion1.19 Beam Deflection ...1.20 Transportation of Logs

Chapter 2. Qualitative Methods ofStudying Differential Models

2.1 Curves of Constant Direction ofMagnetic Needle , . , , , t , ,

7

11111418

2630323438

445155

697381

88

9396

100112119

136

136

6 Contents

2.2 Why Must an Engineer KnowExistence and Uniqueness Theo-~m~ • M3

2.3 A Dynamical Interpretation ofSecond-Order Differential Equa-tion! .. 155

2.4 Conservative Systems in Mechanics 1622.5 Stability of Equilibrium Points

and of Periodic Motion 1762.6 Lyapunov Functions . . . 1842.7 Simple States of Equilibrium. 1892.8 Motion of a Unit-Mass Object

Under the Action of LinearSprings in a Medium with LinearDrag . 195

2.9 Adiabatic Flow of a Perfect GasThrough a Nozzle of VaryingCross Section . . . .. 203

2.10 Higher-Order Points of Equilib-rium . . 211

2.11 Inversion with Respect to a Circleand Homogeneous Coordinates 218

2.12 Flow of a Perfect Gas Through aRotating Tube of Uniform CrossSection . ..... 223

2.13 Isolated Closed Trajectories. 2362.14 Periodic Modes in Electric Circuits 2502.15 Curves Without Contact. . 2602.16 The Topographical System of

Curves. The Contact Curve . .. 2632. t 7 The Divergence of a Vector Field

and Limit Cycles 270

Selected Readings 274

Appendices 275

Appendix 1. Derivatives of Elemen-tary Functions.. 275

Appendix 2. Basic Integrals , , , ., 278

Preface

Differential equations belong to one of themain mathematical concepts. They areequations for finding functions whose derivatives (or differentials) satisly given conditions. The differential equations arrived atin the process of studying a real phenomenonor process are called the differential modelof this phenomenon or process. It is clearthat differential models constitute a particular case of the numerous mathematical modelsthat can be built as a result of studies ofthe world that surrounds us. It must beemphasized that there are different types ofdifferential models. This book considers nonebut models described by what is knownas ordinary differential equations, one characteristic of which is that the unknownfunctions in these equations depend ona single variable.

In constructing ordinary differential models it is important to know the laws ofthe branch of science relating to the natureof the problem being studied. For instance,in mechanics these may be Newton's laws,in the theory of electric circuits Kirchhoff'slaws, "in the theory of chemical reactionrates the law of mass action.

Of course, in practical life we often haveto deal with cases where the laws that "en-

8 Preface

able building a differentia lequation (or several differential equations) are not known,and we must resort to various assumptions(hypotheses) concerning the course of theprocess at small variations of the parameters, the variables. Passage to the limitwill then lead to a differential equation,and if it so happens that the results ofinvestigation of the differential equation asthe mathematical model agree with the experimental data, this will mean that thehypothesis underlying the model reflectsthe true situation.*

When working on this book, I had twogoals in mind. The first was to use examples,rich in content rather than purely illustrative, taken from various fields of knowledgeso as to demonstrate the possibilities ofusing ordinary differential equations ingaining an understanding of the world aboutus. Of course, the examples far from exhaust the scope of problems solvable by ordinary differential equations. They givean idea of the role that ordinary differentialequations play in solving practical problems.

The second goal was to acquaint the read-

• If the reader wishes to know more about mathematical models, he can turn to the fascinating booksby A.N. Tikhonov and D.P. Kostomarov, StoriesAbout Applied Mathematics (Moscow: Nauka,1979) (in Russian), and N.N. Moiseev, Mathematics Stages an Experiment (Mo~CQw: Nauka, t979)(in Ru~ialt),

Preface 9

er with the simplest tools and methodsused in studying ordinary differential equations and characteristic of the qualitativetheory of differential equations. The fact isthat only in rare cases are we able to solvea differential equation in the so-calledclosed form, that is, represent the solutionas a formula that employs a finite numberof the simplest operations involving elementary functions, even when it is knownthat the differential equation has a solution. In other words, we can say that thegreat variety of solutions to differentialequations is such that for their representation in closed form a finite number of analytical operations is insufficient. A similar situation exists in the theory of algebraicequations: while for first- and second-orderalgebraic equations the solutions can alwaysbe easily expressed in terms of radicalsand for third- and fourth-order equationsthe solutions can still be expressed in termsof radicals (although the formulas becomevery complicated), for a general algebraicequation of an order higher than the fourththe solution cannot be expressed in radicals.

To return to differential equations. Ifan infinite series of this or that form is usedto represent the solutions, then the scopeof solvable equations broadens considerably. Unfortunately, it often happens thatthe most essential and interesting properties of the solutions cannot be revealed by

10 Preface

studying the form of the series. More thanthat, even if a differential equation can besolved in closed form, more often than notit is impossible to analyze such a solutionsince the relationship between the variousparameters of the solution often provesto be extremely complicated.

This shows how important it is to developmethods that make it possible to acquirethe data on the various properties of thesolutions without solving the differentialequations themselves. And indeed, suchmethods do exist. They constitute the essence of the qualitative theory of differential equations, which is based on the general theorems regarding the existence anduniqueness of solutions and the continuousdependence of solutions on the initial dataand parameters. The role of existence anduniqueness theorems is partially discussedin Section 2.2. As for the general qualitativetheory of ordinary differential equations,ever since J.H. Poincare and A.M. Lyapunov laid the foundations at the end ofthe 19th century, the theory has been intensively developing and its methods are widely used when studying the world about us.

I am indebted to Professors Yu.S. Bogdanov and M.V. Fedoryuk for the constructive remarks and comments expressed inthe process of preparing the book for publication.

v.V. Amel'kin

Chapter 1

Constructionof Differential Modelsand Their Solutions

1.1 \Vhose Coffee Was Hotter?

When Tom and Dick ordered coffee andcream in a lunch room, they were given bothsimultaneously and proceeded as follows.Tom poured some of his cream into the coffee, covered the cup with a paper napkin,and went to make a phone call. Dick covered his cup with a napkin and poured thesame amount of cream into his coffee onlyafter 10 minutes, when Tom returned .. Thetwo started drinking their coffee at the sametime. Whose was hotter?

We will solve this problem on the naturalassumption that according to the laws ofphysics heat transfer through the surface ofthe table and the paper napkin is much lessthan through the sides of the cups and thatthe temperature of the vapor above the surface of the coffee in the cups equals the temperature of the coffee.

We start by deriving a relationship indicating the time dependence of the temperature of the coffee in Dick's cup before thecream is added.

In accordance with our assumption onthe basis of a law of physics, the amount

12 Differential Equations in Applications

of heat transferred to the a ir from Dick'scup is determined by the formula

T-9dQ=l1'-I-sdt, (1)

where T is the coffee's temperature at timet, e the temperature of the air in the lunchroom, 11 the thermal conductivity of thematerial of the cup, l the thickness of thecup, and s the area of the cup's lateral surface. The amount of heat given off by thecoffee is

dQ = -em dT, (2)

where c is the specific heat capacity of thecoffee, and m the mass (or amount) of coffeein the cup. If we now consider Eqs. (1)and (2) together, we arrive at the followingequation:

T-8ll-l-sdt= -cmdT,

which, after variable separation, can be rewritten as follows:

dT 118T-8 = - lcm dz. (3)

Denoting the initial temperature of the coffeeby To and integrating the differential equation (3), we find that

T=6+(To- 6) exp ( - l~~ t). (4)

This formula is the analytical expressionof the law whereby the temperature of the

Ch. 1. Construction of Differential Models 13

coHee in Dick's cup varied prior to the addition of the cream .• Now let us establish the law whereby thetemperature of the coffee in Dick's cupchanged after Dick poured in cream. Weuse the heat balance equation, which herecan be written as

em (T - en) == elml (en - T1), (5)

where eo is the temperature of the Dick'scoffee with cream at time t, T I the temperature of the cream, cl the specific heat capacity of the cream, and In! the mass ofcream added to the coffee.

Equation (5) yields

8n = Clmt T',+ em T. (6)em+elml em+clml

Bearing in mind formula (4), we can rewrite (6) as follows:

x [S+(To-S)exp( - l~ t)J, (7)

which constitutes the law whereby the temperature of the coffee in Dick's cup variesafter cream is added.

To derive the law for temperature variation of the coffee in Tom's cup we againemploy the heat balance equation, whichnow assumes the form

em (To - 80) = elm l (eo - T t ) , (8)


where 80 is the temperature of the mixture.If we solve (8) for 80 , we get

Then, using Eq. (4) with 80 serving as theinitial temperature and cm + c1nt1 substituted for em, we arrive at the law for thetemperature (8T ) variation of the coffeein Tom's cup as analytically given by thefollowing formula:

Thus, to answer the question posed inthe problem we need only turn to formulas(7) and (9) and carry out the necessary calculations, bearing in mind that C1 ~ 3.9 X103 J/kg·K, C ~ 4.1 X 103 J/kg·K, and'll~O.6V/m·K and assuming, for the sakeof definiteness, that m1 = 2 X 10-2 kg,m = 8 X 10-2 kg, T1 = 20°C, e = 20°C,To = 80 °C, S = 11 X 10-3 m", and l =2 X 10-3 m. The calculations show thatTom's coffee was hotter.

1.2 Steady-State Heat Flow

The reader will recall that a steady-stateheat flow is one in which the object's tern-

Ch, 1. Construction of Differential Models 15

Fig. 1 Fig. 2

perature at each point does not vary withtime.

In problems whose physical content isrelated to the effects of heat flows, an important role is played by the so-called isothermal surfaces. To clarify this statement, letus consider a heat-conducting pipe (Figure1) 20 em in diameter, made of a homogeneous material, and protected by a layer ofmagnesium oxide 10 ern thick. We assumethat the temperature of the pipe is 160°Cand the outer surface of the protective covering has a temperature of 30 oe. It isintuitively clear then that there is a surface,designated by a dashed curve in Figure2, at each point of which the temperatureis the same, say 95 "C. The dashed curve inFigure 2 is known as an isotherm, while the


surface corresponding to this curve:is knownas an isothermal surface. In general, isotherms may have various shapes, depending, for one, on the nonsteady-state natureof the heattflow and on the nonhomogeneityof the material. In the case at hand theisotherms (isothermal surfaces) are represented by concentric circles (cylinders).

We wish to derive the law of temperaturedistribution inside the protective coatingand find the amount of heat released by thepipe over a section 1 m long in the courseof 24 hours, assuming that the thermal conductivity coefficient k is equal to 1.7 X10-4 •

To this end we turn to the Fourier law,according to which the amount of heat released per unit time by an object that is in stable thermal state and whose temperature T ateach point is solely a function of coordinate xcan be found according to the formula

dTQ = -kF (x) dX = const, (10)

where F (x) is the cross-sectional area normalto the direction of heat flow, and k the thermalconductivity coefficient.

The statement of the problem impliesthat F (x) = Ztud; where 1 is the length ofthe pipe (em), and x the radius of the baseof the cylindrical surface lying inside theouter cylinder. Then on the basis of (10)


we get

30 20r Q r dzJ dT = - 0.00017 X2rcl J7'

160 10

T x\ Q \ d;J dr = - 0.0001.7 X 2nl J -~-.

160 10

Integrating (11) and (12) yields

(11)

(12)

160-T130

Hence

In O.1x _ log O.lxIn 2 log 2

T = 591.8 - 431.8 log x.

This formula expresses the law of temperature distribution inside the protectivecoating. We see that the length of the pipeplays no role in this law.

To answer the second question, we turnto Eq. (11). Then for l = 100 em we findthat

Q= 130 X 0.00017 X 2 X 100In2

200n X 130 X 0.000170.69315

and, hence, the amount of heat released bythe pipe in the course of 24 hours is 24 X60 X 60Q = 726 852 J.2-0770


1.3 An Incident in a National Park

While patrolling a national park, two forest rangers found the carcass- of a slain wildboar. Inspection showed that the poacher'sshot had been precise and the boar had diedon the spot. Reasoning that the poachershould return for the kill, the rangers decided to wait for him and hid nearby. Soontwo men appeared, heading directly for thedead boar. When confronted by the rangers,the men denied having anything to do withthe poaching. But by this time the rangershad collected indirect evidence of theirguilt. It only remained to establish theexact time of the kill. This they did byusing the law of heat emission. Let us seewhat reasoning this involved.

According to the law of heat emission,the rate at which an object cools off in airis proportional to the difference betweenthe temperature of the object and that ofthe air,

dx-= -k(x-a)dt '

(13)

where x is the temperature of the object attime t, a the temperature of the air, andk a positive proportionality coefficient.

Solution of the problem lies in an analy..sis of the relationship that results from integrating the differential equation (13).Here one must bear in mind that after the


boar was killed, the temperature of the aircould have remained constant but alsocould have varied with time. In the first caseintegration of the differential equation (13)with variables separable leads to

x-aIn--==-kt, x =1= a, (14)xo-a

where X o is the temperature of the objectat time t = O. Then, if at the time whenthe strangers were confronted by the rangersthe temperature of the carcass, x, was 31°Cand after an hour 29 °C and if when the shotwas fired the temperature of the boar wasx = 37°C and the temperature of the aira = 21°C, we can establish when the shotwas fired by putting t = 0 as the time whenthe strangers were detained. Using thesedata and Eq. (14), we find that

31-21k=ln 29-21 ==ln1.25==O.22314. (15)

Now, substituting this value of k and x ==37 into (14), we get

1 37-21t = - 0.22314 In 31-21

10.22314 In 1.6 == - 2.10630.

In other words, roughly 2 hours and 6 minutes passed between the time the boar waskilled and the time the strangers were detained.

2*


In the second case, that is, when the temperature of the air varies in time, the cooling off of the carcass is expressed by thefollowing nonhomogeneous linear differential equation

:~ + kx=ka(t), (16)

where a (t) is the temperature of the air attime i.

To illustrate one of the methods for determining the time when the boar was killed,let us assume that the temperature of thecarcass was 30°C when the strangers weredetained. Let us also suppose that it isknown that on the day of the kill the temperature of the air dropped by 1 °C everyhour in the afternoon and was 0 °C whenthe carcass was discovered. We will alsoassume that after an hour had passed afterthe discovery the temperature of the carcasswas 25°C and that of the air was downto -1° C. If we now assume that the shotwas fired at t = 0 and that X o = 37°Cat t = 0, we get a (t) = t* - t, wheret = t* is the time when the rangers discovered the carcass.

Integrating Eq. (16), we get

x = (37 - t - k-1) e-k t + t* - t + k-1•

If we now bear in mind that x = 30°Cat t = t* and x = 25°C at t = t* + 1,

ChI 1. Construction of Differential Models 21

the last formula yields

(37 - t* - k-1) exp (-kt*) + k- 1 = 30,

(37 - t* - k-1) exp [- k (t* + 1)]

+ k-1 = 26.

These two equations can be used to derivean equation for k, namely,

(30 - k-1)e-k - 26 + k-1 = O. (17)

We can arrive at the same equation starting from different assumptions. Indeed,suppose that at t = 0 the carcass of the slainboar was found. Then a (t) = -t andwe arrive at the differential equation

dx +dt kx= -kt (18)

(with the initial data X o = 30 at t == 0),from which we must find x as an explicitfunction of t.

Solving Eq. (18), we get

x = (30 - k-1) exp (-kt) - t + Ie-i. (19)

Setting x = 25 and t = 1 in this relationship, we again arrive at Eq. (17), which enables solving the initial problem numerically.

Indeed, as is known, Eq. (17) cannot besolved algebraically for k, But it is easilysolved by numerical methods for finding theroots of transcendental equations, for one,


by Newton's method of approximation. Thismethod, as well as other methods of successive approximations, is a way of using arough estimate of the true value of a rootused to obtain more exact values of theroot. The process can be continued untilthe required accuracy is achieved.

To show how Newton's method is used,we transform Eq. (17) to the form

30k - 1 + (1 - 26k) exp (k) = 0, (20)

and Eq. (19), setting x == 37, to the form

(37k - 1 + kt) exp (kt) - 30k + 1 == O.(21)

Both equations, (20) and (21), are of thetype

(ax + b) exp (Ax) + ex + d = O. (22)

If we denote the left-hand side of Eq. (22)by qJ (x), differentiation with respect to xyields

tp' (x) = (Aax + 'Ab + a) exp (A,x) + c,

qJ" (x) = (A,2ax + 'A2b + 2Aa) exp (Ax).

Then according to Newton's method forfinding a root of Eq. (22), if for the ithapproximation Xi we have the inequality

tp (Xi) cP" (Xi) > 0,

the next approximation, Xt+l' can be found

via the formula

q> (z i)Xi+l = Xi - cp' (Xi) •

To directly calculate the root (to within,say, one part in a million), we compilethe following program using BASIC*:

10 CLS:PRINT "Solution of equation by New-ton's method"

i5 INPUT "lambda==", L;20 INPUT "a=", A: INPUT "b==", B;30 INPUT "c=", C: INPUT "d==", D40 INPUT "approximate value of root==", X50 PRINT "X", "f", "f"', "f""100 E == EXP (L*X)110 F == (A*X + B)*E + C·X + D120 F1 == (L*(A*X + B) + A)*E + C130 F2 == L* (L* (A*X + B) + 2*A)*E150 PRINT X, F, F1, F2'151 IF F == 0 THEN END155 IF F1 = 0 THEN PRINT "Newton's method

is divergent": END170 IF F*F2 < 0 THEN PHINT "Newton's

method is divergent": END190 X = X - F/Fi200 GOTO 100

In this program X, f, f' and f" stand forkn , cp (kn) , <p' (kn ) , and <p" (kn,) , respectively.

* Editor's note. Some BASIC lines, e.g, line10, have been split due to the printed format;they should be input as one line.

After starting the program enter the requested values of the coefficients of the equation and the initial value of the root. Theresults can be listed in a table of the approximate values of the root and the respectivevalues of the function and its first andsecond derivatives.. Employing this general procedure, let usturn to Eq. (20). Differentiating its left-handside ~ (k) with respect to k, we arrive at

q/ (k) == 30 - (25 + 26k) exp (k).

I t can then easily be verified that (f (0) =0, cp (1) < 0, and q/ (0) > O. Thus, thefunction {p increases in a small neighborhoodof the origin and then decreases to a negative value at k = 1. This implies that in theinterval (0, 1) there is a root of the equation(j) (k) == 0. To find this root we run the program. Below we give the protocol of theprogram:

Solution of equation by Newton's methodlambda>- 1

a == -26 b = 1 c = 30 d = - 1

Approximate value of root = 0.5

X f f' f".5 -5.784655 -32.65141 -105.5182.3~2836 -1.525956 -16.11805 -82.02506.2281622 -.3514252 -8.859806 -71.52333.1884971 -5.519438E-02 -6.103379 -67.49665•t 701~3\} -~.7480l3E-03 -5,~970~f -QQ,60768

Chi 1. Construction of Differential Models 25

.178954 -7.629395E-06 -5.46373 -66.55884

.1789526 -4.768372E-07 -5.463642 -66.5587

.1789525 0 -5.463635 -66.5587OK

In this protocol X, I, f', and f" stand foru; {P (kll ) , £P' (kn ) , and <p" (kn ) , respectively.

The final step in solving the problem consists in substituting the calculated valuek6 ~ k ~ 0.178952 into Eq. (21) and solving the latter for t (the time when the wildboar was killed). To employ the abovescheme, we denote the left-hand side of Eq.(21) by g (t). Then, selecting -1 for thevalue of to and bearing in mind that inthis case a = k, b == 37k - 1, c == 0, d =-30k + 1, and Iv ==k, we can find the timewhen the boar was killed. To this endwe find the coefficients band d, which proveto be equal to 5.621243 and -4.368575, respectively, and then running the above program, find the sought time t. The protocol of the program is given below:

Solution of equation by Newton's method

lambda = 0.1789525a = 0.1789525 b = 5.621243 c = 0d = -4.368575Approximate value of root = -1

X I I' I"--1 .181972 .9639622 .1992802-1.188775 3.506184E-03 ,9270549 .1917861


Fig. 3

-1.192557 1.430512E-06 .9263298 .1916388-1.192559 0 .9263295 .1916387OK

In this protocol X, f, f", and f" stand fortn' g (t n ) , g' (t n ) , and g" (t n ) , respectively.

These results imply that the boar waskilled approximately 1 hour and 12 minutesbefore the rangers discovered the carcass.

t.4 Liquid Flow Out of Vessels.The Water Clock

The two problems that we now discuss illustrate the relationship between the physicalcontent of a problem and geometry. But firstlet us examine some general theoretical conclusions.

We take a vessel (Figure 3) whose horizontal cross section has an area that is afunction of the distance from the bottomof the vessel to the cross section. Suppose

(23)


that initially at time t = 0 the level of theliquid in the vessel is at a height of hmeters. We will also suppose that the areaof the vessel's cross section at height xis denoted by 8 (x) and that the area of theopening in the bottom of the vessel is s.

As is known, the rate v at which the liquidflows out of the vessel at the moment when theliquid's level is at height x is given by the for-mula v = k V2gx, where g = 9.8 m/s2 ,

and k is the rate constant of the outflow process.

In the course of an infinitesimal timeinterval dt the outflow of the liquid can beassumed uniform, whereby during dt acolumn of liquid with a height of v dtand a cross-sectional area of s will flow outof the vessel, which causes the level of theliquid to change by -dx (the "minus" because the level lowers).

The above reasoning leads us to the following differential equation

ksV2gx dt = -8 (x) dx,

which can be rewritten as

dt = _ S(x) dz.ks y2gx

Let us now solve the following problem.A cylindrical vessel with a vertical axissix meters high and four meters in diameterhas a circular opening in the bottom. The

26 Differential Equations in Applica tions

Fig. 3

-1.192557 1.430512E-06 .9263298 .1916388--1.192559 0 .9263295 .1916387OK

In this protocol X, f, f', and f" stand fortn, g (t n ) , g' (tn ) , and g" (t n), respectively.

These results imply that the boar waskilled approximately 1 hour and 12 minutesbefore the rangers discovered the carcass.

1.4 Liquid Flow Out of Vessels.The Water Clock

The two problems that we now discuss illustrate the relationship between the physicalcontent of a problem and geometry. But firstlet us examine some general theoretical conclusions.

We take a vessel (Figure 3) whose horizontal cross section has an area that is afunction of the distance from the bottomof the vessel to the cross section, Suppose

(23)


that initially at time t = 0 the level of theliquid in the vessel is at a height of hmeters. We will also suppose that the areaof the vessel's cross section at height xis denoted by S (x) and that the area of theopening in the bottom of the vessel is s.

As is known, the rate v at which the liquidflows out 0/ the vessel at the moment when theliquid's level is at height x is given by the for-mula v = k V2gx, where g = 9.8 m/s2 ,

and k is the rate constant of the outflow process.

In the course of an infinitesimal timeinterval dt the outflow of the liquid can beassumed uniform, whereby during dt acolumn of liquid with a height of v dtand a cross-sectional area of s will flow outof the vessel, which causes the level of theliquid to change by -dx (the "minus" because the level lowers).

The above reasoning leads us to the following differential equation

ksV2gx dt = -8 (x) dx,


dt = _ S (x) dx.ks y2gx

Let us now solve the following problem.A cylindrical vessel with a vertical axissix meters high and four meters in diameterhas a circular opening in the bottom. The


radius of this opening is 1/12 m. Find howthe level of water in the vessel depends ontime t and the time it takes all the wate-r toflow out. T

By hypothesis, S (x) = 43t and s =1/144. Since for water k = 0.6, Eq. (23)assumes the form

dt = - 217.152 dyx x,

Integrating this differential equation yields

t =434.304 [V6 - ¥Xl, O~x~6,

which is the sought dependence of the levelof water in the vessel on time t. If we putx = 0 in the last formula, we find that ittakes approximately 18 minutes for all thewater to flow out of the vessel.

Now a second problem. An ancient waterclock consists of a bowl with a small holein the bottom through which water flowsout of the bowl (Figure 4). Such clocks wereused in ancient Greek and Roman courts totime the lawyers' speeches, so as to avoidprolonged speeches. We wish to determinethe shape of the water clock that would ensure that the water level lowered at a constant rate.

This problem can easily be solved viaEq. (23). 'Ve rewrite this equation in theform

Vx==- - S(x) ~ (24)ks y2g dt •


Fig. 4

Precisely, if we suppose that the bowl hasthe shape of a surface of revolution, in accordance with the notations used in Figure4, Eq. (24) yields the following result:

V- nr2X= - a

ks V2g ,(25)

where a == Vx == rlx/dt is the projection onthe x axis of the rate of motion of thewater's free surface, which is constant byhypothesis. Squaring both sides of Eq.(25), we arrive at the equation

x = cr", (26)

with c = a2n2/2gk2s2• The latter means that

the sought shape of the water clock is obtained by rotating curve (26) about the xaxis.


1.5 Effectiveness of Advertising

Suppose that a retail chain is selling commodities of a certain type, say B, aboutwhich only x buyers out of N potential buyers know at time t. Let us also assume thatto speed up sales the chain has placed promotion rnaterials in the local TV and radionetwork. All further information about thecommodities is distributed among the buyers via personal contact between them. \Vemay assume with a high probability thatafter the TV and radio network have released the information about B, the rate ofchange in the number of persons knowingabout B is proportional both to the numberof buyers knowing about B and to the number of buyers not knowing about B.

If we suppose that time is reckoned fromthe moment when the promotion materialsand advertisements were released andN /"( persons have learned about the commodities, we arrive at the following differential equation:

dxdj=kx (N - x), (27)

with the initial condition that x == .TVI,,?at t = O. In this equation k is a positiveproportionality factor. Integrating, we findthat

1 xN 1n N-x =kt+C.


N

o t

Fig. 5

Assuming that NC == C1, we arrive at theequation

_x__=AeN k tN-x '

(28)N

with A = eel. When solved for x, thisequa tion yields

AeN k tx==N-~--

AeN k t +1

with P = 1/A.In economics, Eq. (28) is commonly

known as the logistic-curve equation.If we allow for the initial condition, Eq.

(28) becomes

x = ------.--N1+(y_1)e-N k t •

Figure 5 provides a schematic of a logistic curve for "( == 2. In conclusion we note


that the problem of dissemination of technological innovations can also be reducedto Eq. (27).

1.6 Supply and Demand

As is known, supply and demand. constituteeconomic categories in a commodity economy, categories that emerge and function onthe market, that is, in the sphere of trade.The first ca tegory represents the commodities that exist on the market or can be delivered to it, while the second represents thedemand for commodities on the market.One of the main laws of a market economyis the law of supply and demand, whichcan be formulated simply by saying thatfor each commodity some price must existthat will cause the supply and the demandto be just equal. Such a price establishes an"equilibrium" on the market.

Let us consider the following problem.Suppose that in the course of a (relativelylong) time interval a farmer sells his produce(say, apples) on the market, and that hedoes this immediately after the apple harvest has been taken in and then once eachfollowing week (i.e. with a week's interval).The farmer's stock of apples being fixed after he has collected his harvest, the week'ssupply will depend on both the expectedprice of apples in the coming week and theexpected change in price in the following

Ch. 1. Const.ruction of Differential Models 33

weeks. If it is assumed that in the comingweek the price of apples will fall and in thefollowing weeks it will grow, then supplywill be restrained provided that the expected rise in price exceeds storage costs. Inthese conditions the greater the expected risein price in the following weeks, the lowerthe supply of apples on the market. On theother hand, if the price of apples in thecoming week is high and then a fall is expected in the following weeks, then thegreater the expected fall in price in the following weeks, the higher the supply of apples on the market in the coming week.

If we denote the price of apples in thecoming week by p and the time derivativeof price (the tendency of price formation)by p', both supply and demand are functions of these quantities. As practice hasshown, depending on various factors thesupply and demand of a commodity may berepresented by different functions of priceand tendency of price formation. For example, one function is given by a linear dependence expressed mathematically in theform y = ap' + bp + c, where a, b, andc are real numbers (constants). Say, inour example, the price of apples in thecoming week is taken at 1 rouble for 1 kg,in t weeks it was p (t) roubles for 1 kg,and the supply s and demand q are givenby the functionss = 44p' + 2p - 1, q = 4p' - 2p + 39.3-0770


Then, for the supply and demand to be inequilibrium, we must require that

44p' + 2p - 1 == 4p' - 2p + 39.

This condition leads us to the differentialequation

dpp-10 = -10dt.

Integration yields p == Ce- 1ot + 10. If weallow for the initial condition that p = 1at t = 0, the equilibrium price is given bythe formula

p == _ge-1ot + 10. (29)

Thus, if we want the supply and demand tobe in equilibrium all the time, the pricemust vary according to formula (29).

1.7 Chemical Reactions

A chemical equation shows how the interaction of substances produces a new substance. Take, for instance, the equation

2H 2 + O2 -+ 2H 20.

It shows how as a result of the interactionof two molecules of hydrogen and one molecule of oxygen two molecules of water areformed.


Generally a chemical reaction can be written in the form

aA -1- bB + cC -1

+pP +~lnllf + nN

where A, B, C, are molecules of the interacting substances, M, N, P, . _ molecules of the reaction products, and the constants a, b, C, • _, m, n, p,. positiveintegers that stand for the number of molecules participating in the reaction.

The rate at which a new substance isformed in a reaction is called the reactionrate, and the active mass or concentrationof a reacting substance is given by the number of moles of this substance per unit volume.

One of the basic laws of the theory ofchemical reaction rates is the law of massaction, according to which the rate of a chemical reaction proceeding at a constant temperature is proportional to the product of theconcentrations of the substances taking partin the reaction at a given moment.

Let us solve the following problem. Twoliquid chemical substances A and B occupying a volume of 10 and 20 liters, respectively, form as a result of a chemical reactiona new liquid chemical substance C. Assuming that the temperature does not changein the process of the reaction and that everytwo volumes of substance A and one vol-

3*

ume of substance B form three volumes ofsubstance C, find the amount of substanceC at an arbitrary moment of time t if ittakes 20 min to form 6 liters of C.

Let us denote by x the volume (in liters)of substance C that has formed by the timet (in hours). By hypothesis, by that time2x/3 liters of substance A and x/3 liters ofsubstance B will have reacted. This meansthat we are left with 10 - 2x/3 liters of Aand 20 - x/3 liters of B. Thus, in accordance with the law of mass action, we arriveat the following differential equation

~; = K ( 10 - 2; )(20 - ~ )


dx<it == k (15-x) (60- x),

where k is the proportionality factor (k ==2K/9). We must also bear in mind that sinceinitially (t = 0) there was no substanceC, we may assume that x = 0 at t = O. Asfor the moment t = 1/3, we have x = 6.

Thus, the solution of the initial problemhas been reduced to the solution of a socalled boundary value problem:

dx(ft=k (15 - x) (60 - x), x (0) = 0,

x (1/3) == 6.


To solve it, we first integrate the differential equation and allow for the initial condition x (0) == O. As a result we arriveat the relationship

60-x 4 k___ ~- e45 t15-x - ·

Since x == 6 at t == 1/3, substituting thesevalues into the relationship yields e16h ==

3/2. Hence,

6O-x15-x ==4 (et 5h)3t ~ 4 (3/2)3t,

that is,

1- (2/3)3t

1- (1/4) (2/3)3tX ::= 15------

This gives the amount of substance Cthat has been formed in the reaction bytime t.

A remark is in order. From practicalconsidera tions it is clear that only a finitevolume of substance C can be formed when10 liters of A interact chemically with 20liters of B. However, a formal study ofthe above relationship between x and tshows that for a finite t, namely at (2/3)3t ==

4, the variable x becomes infinite. But thisfact does not contradict practical considerations because it is realized only for anegative value of t, whi.le the chemical reaction is considered only for t nonnegative.


t.8 Differential Models in Ecology

Ecology studies the interaction of DIan and,in general, living organisms with the environment. The basic object in ecology is theevolution of populations. Below we describedifferential models of populations that dealwith their reproduction or extinction andwith the coexistence of various species ofanimals in the predator vs. prey situation. *

Let x (t) be the number of individualsin a population at time t. If A is the numberof individuals in the population that areborn per unit time and B the number ofindividuals that die off per unit time, thenthere is reason to say that the rate at whichx varies in time is given by the formuladx(It=A-B. (30)

The problem consists in finding the dependence of A and B on x. The simplest situation is the one in which

A = ax, B = bx, (31)

where a and b are the coefficients of birthsand deaths of individuals per unit time,respectively. Allowing for (31), we can rewrite the differential equation (30) asdx(It==(a-b)x. (32)

* For more detail see J.D. Murray, "Some simplemathematical models in ecology," Math. Spectrum16, No.2: 48-54 (1983/84).

Ch. 1. Construction of Differential Models 3!l

Assuming that at t = to the number of individuals in the population is x == X oand solving Eq. (32), we get

x (t) = Xo exp [(a - b) (t - to»).

This implies that if a > b, then the numberof individuals x tends to infinity as t -+ 00,

but if a < b, then x -+ 0 as t -+ 00 andthe population becomes extinct.

Although the above model is a simplifiedone, it still reflects the actual situation insome cases. However, practically all models describing real phenomena and processes are nonlinear, and instead of the differential equation (32) we are forced to consider an equation of the typedxdt===!(x),

where f (x) is a nonlinear function, say theequationdxdt=== f (x) = ax - bx«,

where

a> 0, b > O.Assuming that x = X o at t = to and solvingthe last equation, we get

x (t) = xoa/b . (33)xo+ (alb -xo) exp [- a (t-to}) •

We see that as t -+ 00 the number of individuals in the population, x (t), tends to

aa: 75>.:co

~-~~~~------~--~--

i

----------a-----F<xo

OL.-------=-----~

Fig. 6

a/b. Two cases are possible here: alb> X oand alb < X O• The difference between thetwo is clearly seen in Figure 6. Note thatformula (33) describes, for one thing, thepopulations of fruit pests and some typesof bacteria.

If we consider several coexisting species,say, big and small fish, where the small fishserve as prey for the big, then by setting updifferential equations for each species wearrive at a system of equations

dXidt == i. (X., Xn), i = 1, 2, n.

Let us study in greater detail the two-species predator vs, prey model, first introduced by the Italian mathematician VitoVolterra (1860-1940) to explain the oscillations in catch volumes in the Adriatic Sea,which have the same period but differ inphase.


Let x be the number of big fish (predators)that feed on small fish (prey), whose number we denote by y. Then the number ofpredator fish will grow as long as they havesufficient food, that is, prey fish. Finally, asituation will emerge in which there willnot be enough food and the number of bigfish will diminish. As a result, starting froma certain moment the number of small fishbegins to increase. This, in turn, assists anew growth in the number of the big species, and the cycle is repeated. The modelconstructed by Volterra has the form

dxdt= -ax+bxy,

dydt==cx- dxy,

(34)

(35)

where a, b, C, and d are positive constants.In Eq. (34) for the big fish the term bxy

reflects the dependence of the increase inthe number of big fish on the number ofsmall fish, while in Eq. (35) the term-dxy expresses the decrease in the numberof small fish as a function of the number ofbig fish.

To make the study of these two equationsmore convenient we introduce dimensionless variables:

U (T) =!£ x, v (t) =.!!- y, 1: = ct, a, = a/c.c a

As a result the differential equations (34)and (35) assume the formu' == au (v - 1), o' == v (1 - u), (36)with CJ., positive and the prime standing fordifferentiation with respect to 'to

Let us assume that at time T == To thenumber of individuals of both species isknown, that is,u ('to) == U O, v (To) = Vo. (37)

Note that we are interested only in positivesolutions. Let us establish the relationship between u and v. To this end we dividethe first equation in system (36) by the second and integrate the resulting differentialequation. We get

a» + u - In vau = a,V o + U o - In v~uo

=H,

where H is a constant determined by theinitial conditions (37) and parameter a,

Figure 7 depicts the dependence of uon v for different values of H. We see thatthe (u, v)-plane contains only closed curves.Let us now assume that the initial values Uo and Vo are specified by point Aon the trajectory that corresponds to thevalue H = H:l. Since U o > 1 and Vo < 1,the first equation in (36) shows that at firstvariable u decreases. The same is true ofvariable v. Then, as u approaches a valueclose to unity, v' vanishes, which is followed by a prolonged period of time 't in the


u

o

Fig. 7

v

a

Fig. 8

course of which variable v increases. Whenv becomes equal to unity, u' vanishes andvariable u begins to increase. Thus, bothu and v traverse a closed trajectory. Thismeans that the solutions are functions periodic in time. The variable u does not achieve

its maximum when v does, that is, oscillations in the populations occur in different phases. A typical graph of u and v asfunctions of time is shown in Figure 8 (forthe case where Vo > 1 and Uo < 1).

In conclusion we note that the study ofcommunities that interact in a more complex manner provides results that are moreinteresting from the practical standpointthan those obtained above. For example, iftwo populations fight for the same source offood (the third population), it can be demonstrated that one of them will becomeextinct. Clearly, if this is the third population (the source of food), then the other twowill also become extinct.

1.9 A Problem from theMathematical Theory of Epidemics

Let us consider a differential model encountered in the theory of epidemics. Supposethat a population consisting of N individuals is split into three groups. The first include individuals that are susceptible to acertain disease but are healthy. We denotethe number of such individuals at timet by S (t). The second group incorporatesindividuals that are infected, that is, theyare ill and serve as a source of infection.We denote the number of such idividualsin the population at time t by I (t). Finally, the third group consists of individuals


who are healthy and immune to the disease. We denote the numher of such individuals at time t by R (t). Thus,

S (t) + I (t) + R (t) == N. (~~8)

Let us also assume that when the numberof infected individuals exceeds a certainfixed number 1*, the rate of change in thenumber of individuals susceptible to thedisease is proportional to the number ofsuch individuals. As for the rate of changein the number of infected individuals thateventually recover, we assume that it isproportional to the number of infected individuals. Clearly, these assumptions simplify matters and in a number of cases theyreflect the real situation. Because of thefirst assumption, we suppose that when thenumber of infected individuals I (t) isgreater than 1*, they can infect the individuals susceptible to the disease. This meansthat we have allowed for the fact that theinfected individuals have been isolated fora certain time interval (as a result of quarantine or because they have been far fromindividuals susceptible to the disease).We, therefore, arrive at the following differential equation

dS == { -as if I (t) > 1*, (39)dt 0 ifI(t)~I*.

Now, since each individual susceptibleto the disease eventually falls ill and be-


comes a source of infection, the rate ofchange in the number of infected individualsis the difference, per unit time, between thenewly infected individuals and those thatare getting well. lienee,

~ == { as- ~I if I (t) > 1*, (40)dt -~I if I (t) ~ 1*.

We will call the proportionality factorsa and ~ the coefficients of illness and recovery.

Finally, the rate of change of the numberof recovering individuals is given by theequation dR/dt = ~I.

For the solutions of the respective equations to be unique, we must fix the initialconditions. For the sake of simplicity weassume that at time t = 0 no individualsin the population are immune to the disease,that is, R (0) = 0, and that initially thenumber of infected individuals was I (0).Next we assume that the illness and recovery coefficients are equal, that is, a = ~

(the reader is advised to study the casewhere a =t= ~). Hence, we find ourselves ina situation in which we must consider twocases.

Case 1. I (0)~ 1*. With the passage oftime the individuals in the population willnot become infected because in this casedS/dt = 0, and, hence, in accordance withEq. (38) and the condition R (0) = 0,we have an equation valid for all values


N ----------------------t----.......--------S(t}

-------- -------- ---R(t)Itt)

tFig. 9

of t:S (t) = S (0) == N - I (0).

The case considered here corresponds to thesituation when a fairly large number of infected individuals are placed in quarantine. Equation (40) then leads us to the following differential equationdIdt=-aI.

This means that I (t) == I (0) e-a t and,hence,

R (t) == N - S (t) - I (t)= I (0) [1 - e-atl.

Figure 9 provides diagrams that illustratethe changes with the passage of time in the

number of individuals in each of the threegroups.

Case 2. I (0) > 1*. In this case there mustexist a time interval 0~ t < 11 in whichI (t) > 1* for all values of t, since by itsvery meaning the function I == I (t) iscontinuous. This implies that for all t'sbelonging to the interval [0, Tl the diseasespreads to the individuals susceptible to it.Equation (39), therefore, implies that

S (t) = S (O)e -at

for 0~ t < T. Substituting this into Eq.(40), we arrive at the differential equation

dIdT+al =as (0) e- a t • (41)

If we now multiply both sides of Eq. (41)by eat, we get the equation

d(It (lea t ) =as (0).

Hence, lea t = as (0) t + C and, therefore, the set of all solutions to Eq. (41) isgiven by the formula

I (t) = Ce-a t + as (0) te-at. (42)

Assuming that t = 0, we get C = I (0) and,hence, Eq. (42) takes the form

I (t) = [I (0) + as (0) tl e-a t , (43)

with O~ t < T.


We devote our further investigations tothe problem of finding the specific value ofT and the moment of time tmax at which thenumber of infected individuals provesmaximal.

The answer to the first question is important because starting from T the individualssusceptible to the disease cease to becomeinfected. If we turn to Eq. (43), the aforesaid implies that at t = T its right-handside assumes the value 1*, that is,

1* = [I (0) + as (0) T]e-a T •

But

S (T) = lim S (t) = S (00)t-+oo

(44)

is the number of individuals that are susceptible to the disease and yet avoid fallingill; for such individuals the following chainof equalities holds true:

S (T) = S (00) = S (0) e-a T •

Hence,

T - ..!-. In~ (45)- a S (00) •

Thus, if we can point to a definite valueof S (00), we can use Eq. (45) to predict thetime when the epidemic will stop. Substituting T given by (45) into Eq. (44),. we arriveat the equation

1* = [I (0)+8 (0) In :/~) ] ~\~; 1

4-0770


or1* _ I (0) S (0)

8(00) -8(0) +lnS(oo) ,

which can be rewritten in the form

1* I (0)S (00) +In S (00) = S (0) +In S (0). (46)

Since 1* and all the terms on the right-handside of (46) are known, we can use this equation to determine S (00).

To answer the second question, we turnto Eq. (43). In accordance with the posedquestion, this equation yields

~ = [as (0) - aI (0) - a2S (0) t] e-a t =0.

The moment in time at which I attains itsmaximal value is given by the following formula:

1 [ 1 (0) ]tm ax = a 1 - S (0) •

If we now substitute this value into Eq. (43),we get

I max = S (0) exp {- [1- I (0)/8 (O)])

= S (t max) .

This relationship shows, for one, that attime tm ax the number of individuals susceptible to the disease coincides with thenumber of infected individuals.


N ---------------------

T

~........--------s (t)

!(t)

imax

- -------- - - - - - -- - - -----R(l)

Fig. 1.0

But if t > T, the individuals susceptibleto the disease cease to be infected and

I (t) = I*e-rx(t-T).

Figure 10 gives a rough sketch of the diagrams that reflect the changes with the passage of time of the number of individualsin each of the three groups considered.

t. to The Pursuit Curve

Let us examine an example in which differential equations are used to choose a cor...rect strategy in pursuit problems.

••


o~-----------

Fig. 11

We assume that a destroyer is in pursuitof a submarine in a dense fog. At a certainmoment in time the fog lifts and the submarine is spotted floating on the surfacethree miles from the destroyer. The destroyer's speed is twice the submarine's speed.We wish to find the trajectory (the pursuitcurve) that the destroyer must follow topass exactly over the submarine if the latterimmediately dives after being detectedand proceeds at top speed and in a straightline in an unknown direction.

To solve the problem we first introducepolar coordinates, rand 8, in such a mannerthat the pole 0 is located at the point atwhich the submarine was located when discovered and that the point at which thedestroyer was located when the submarinewas discovered lies on the polar axis r(Figure 11). Further reasoning is based on


the following considerations. First, the destroyer must take up a position in which itwill be at the same distance from pole 0as the submarine, Then it must move abouto in such a way that both moving objectsremain at the same distance from point 0 allthe time. Only in this case will the destroyereventually pass over the submarine whencircling pole O. The aforesaid implies thatthe destroyer must first go straight to pointo until it finds itself at the same distancex from 0 as the submarine.

Obviously, distance x can be found eitherfrom the equationx 3-%v=~

or from the equation

x _ 3+.1:v-~,

where v is the submarine's speed and 2vthe destroyer's speed. Solving these equations, we find that this distance is either onemile or three miles.

Now, if the two still have not met, thedestroyer must circle pole 0 (clockwise orcounterclockwise) and head away from thepole at a speed equal to that of the submarine, v. Let us decompose the destroyer's velocity vector (of length 2v) into two components, the radial component V r and the tangential component Vt (Figure 11).


The radial component is the speed atwhich the destroyer moves away from pole0, that is

drv, = (It,

while the tangential component is the linear speed at which the destroyer circlesthe pole. The latter component, as is known,is equal to the product of the angular velocity de/dt and radius r, that is,

dOvt=rdt ·

But since Vr must be equal to v, we have

vt = Y(2V)2_V2 = V3v.

Thus, the solution of the initial problemis reduced to the solution of a system of twodifferential equations,

dr de ,/-dt=v, r dt = v 3v,

which, in turn, can be reduced to a singleequation, drlr == del va, by excluding thevariable t.

Solving the last differential equation, wefind that

r=Ce9/V S,

with C an arbitrary constant.


If we now allow for the fact that destroyerhegins its motion about pole 0 starting fromthe polar axis r at a distance of x miles awayfrom 0, that is, r = 1 at e = 0 and r = 3 ate = -n, we conclude that in the first caseC === 1 and in the second C = 3enlVS: Thus,to fulfill its mission, the destroyer mustmove two or six miles along a straight linetoward the point where the submarine wasdiscovered and then move in the spiralr = e9/~3 or the spiral r === 3e(e+1()/~

1. t t Combat Models

During the First World War the Britishengineer and mathematician FrederickW. Lanchester (1868-1946) constructed several mathematical models of air battles.Later these models were generalized so as todescribe battles involving regular troops orguerilla forces or the two simultaneously.Below we consider these three models.

Suppose that two opposing forces, xand y, are in combat. We denote the number of personnel at time t measured in daysstarting from the first day of combat operations by x (t) and y (t), respectively. It isthe number of personnel that will play adecisive part in the construction of thesemodels, since it is difficult, practicallyspeaking, to specify the criteria that wouldallow taking into account, when comparingthe opposing sides, not only the number of


personnel but also combat readiness, levelof military equipment, preparedness andexperience of the officers, morale, and agreat many other factors.

We also assume that x (t) and y (t) changecontinuously and, more than that, aredifferentiable as functions of time. Of course,these assumptions simplify the real situation because x (t) and y (t) are integers.But at the same time it is clear that if thenumber of personnel on each side is great,an increase by one or two persons will havean infinitesimal effect from the practicalstandpoint if compared to the effect produced by the entire force. Therefore, we mayassume that during small time intervalsthe change in the number of personnel isalso small (and does not constitute an integral number). The above reasoning is, ofcourse, insufficient to specify concrete formulas for x (t) and y (t) as functions of t, butwe can already point to a number of factorsthat enable describing the rate of changein the number of personnel on the twosides. Precisely, we denote byOLR the rateat which side x suffers losses from diseaseand other factors not related directly tocombat operations, and by CLR the rateat which it suffers losses directly in combatoperations involving side y. Finally by RRwe denote the rate at which reinforcementsare supplied to side x. It is then clear thatthe total rate of change of x (t) is given by


the equation

d:?) =-(OLR+CLR)+RR. (47)

A similar equation holds true for y (t).The problem now is to find the appropriateformulas for the quantities OLR, CLR,and RR and then examine the resultingdifferential equations. The results shouldindicate the probable winner.

We use the following notation: a, b,c, d, g, and h are nonnegative constantscharacterizing the effect of various factorson personnel losses on both sides (x and Y),P (t) and Q (t) are terms that allow for thepossibility of reinforcements being suppliedto x and y in the course of one day, andXo and Yo are the personnel in x and y priorto combat. We are now ready to set up thethree combat models suggested by Lanchester. * The first refers to combat operations involving regular troops:

d~~t) = -ax (t)-by (t) +P (t),

d~~t) =-cx(t)-dy(t)+Q(t).

* Examples of combat operations are given in thearticle by C.S. Coleman, "Combat models" (seeDifferential Equation Models, eds. M. Braun,C.S. Coleman, and D.A. Drew, New York: Springer, 1983, pp. 109-131). The article shows howthe examples agree with the models considered.


In what follows we call this system the Atype differential system (or simply the Atype system).

The second model, specified by the equations

dx (t)"""d"t = - ax (t) - gx (t) Y (t) +P (t),

dY~t: =-dy(t)-hx(t)y(t)+Q(t)

describes combat operations involving onlyguerilla forces. We will call this system theB-type system. Finally, the third model,which we call the C-type system, has theform

dXd~t) = - ax (t) - gx (t) Y (t) +P (t),

dyd(:)

= -~ ex (t) - dy (t) +Q(t)

and describes combat operations involvingboth regular troops and guerilla forces.

Each of these differential equationsdescribes the rate of change in the numberof personnel on the opposing sides as a function of various factors and has the form(47). Losses in personnel that are not directly related to combat operations and are determined by the terms -ax (t) and-dy (t) make it possible to describe theconstant fractional loss rates (in the absenceof combat operations and reinforcements)


via the equations

1 d% 1 dy----a ----dx dt - 'y dt - •

If the Lanchester models contain onlyterms corresponding to reinforcements andlosses not associated with combat operations, this means that there are no combat operations, while the presence of theterms -by (t), -ex (t), -gx (t) Y (t), and-hx (t) y (t) means that combat operationstake place.

In considering the A-type system, weassume, first, that each side is within therange of the fire weapons of the other and,second, that only the personnel directlyinvolved in combat come under fire. Undersuch assumptions Lanchester introduced theterm -by (t) for the regular troops of sidex to reflect combat losses. The factor bcharacterizes the effectiveness of side y incombat. Thus, the equation1 dx

- -==-bY dt

shows that constant b measures the averageeffectiveness of each man on side y. Thesame interpretation can be given to the term-cx (t). Of course, there is no simple wayin which we can calculate the effectivenesscoefficients band c. One way is to writethese coefficients in the form

b = rTjPII' C = rxpx' (48)


where ry and rx are the coefficients of firepower of sides y and x, respectively, and pyand P:x are the probabilities that each shotfired by Bides y and x, respectively, provesaccurate.

We note further that the terms corresponding to combat losses in the A-type systemare linear, whereas in the B-type systemthey are nonlinear. The explanation lies inthe following. Suppose that the guerillaforces amounting to x (t) personnel occupy acertain territory R and remain undetectedby the opposing side. And although the latter controls this territory by firepower, itcannot know the effectiveness of its actions. It is also highly probable that thelosses suffered by the guerilla forces x are,on the one hand, proportional to the number of personnel x (t) on R and, on theother, to the number y (t) of personnel ofthe opposing side. Hence, the term corresponding to the losses suffered by the guerilla forces x has the form -gx (t) y (t), wherethe coefficient reflecting the effectiveness ofcombat operations of side y is, in general,more difficult to estimate than the coeffiient b in the first relationship in (48).Nevertheless, to find g we can use the firepower coefficient ry and also allow for theideas expounded by Lanchester, accordingto which the probability of an accurateshot fired by side y is directly proportionalto the so-called territorial effectiveness A ry


of a single shot fired by y and inverselyproportional to the area A x of the territoryR occupied by side x. Here by A ry wedenote the area occupied by a single guerilla fighter. Thus, with a high probabilitywe can assume that the formulas forfinding g and hare

(49)

Below we discuss in greater detail each ofthe three differential models.

Case A (differential systems of the A-typeand the quadratic law). Let us assume thatthe regular troops of the two opposingsides are in combat in the simple situationin which the losses not associated directlywith combat operations are nil. If, in addition, neither side receives any reinforcement, the mathematical model is reducedto the following differential system:

dz dy 50di= -by, dt= -ex. ( )

Dividing the second equation by the first,we find that

:~ = ~; . (51)

Integrating, we arrive at the following relationship:

b [y2 (t) - y:l = e [x2 (t) - x:l. (52)


Fig. f2

This relationship explains why system (50)corresponds to a model with a quadraticlaw. If by K we denote the constant quantity by: - ex:, the equation

by2 - cx2 = K (53)

obtained from Eq. (52) specifies a hyperbola(or a pair of straight lines if K = 0) andwe can classify system (50) with great precision. We can call it a differential systemwith a hyperbolic law.

Figure 12 depicts the hyperbolas for different values of K; for obvious reasons weconsider only the first quadrant (x ~ 0,y ~ 0). The arrows on the curves point in


the direction in which the number of personnel changes with passage of time.

To answer the question of who wins inthe constructed model (50), we agree to saythat side y (or x) wins if it is the first towipe out the other side x (or y). For example, in our case the winner is side y ifK > 0 since in accordance with Eq. (53)the variable y can never vanish, while thevariable x vanishes at y (t) = Y K/b. Thus,for y to win a victory, it must strive for asituation in which K is .positive, that is,

(54)

Using (48), we can rewrite (54) in the form

(55)

The left-hand side of (55) demonstrates thatchanges in the personnel ratio yolxo givean advantage to one side, in accordancewith the quadratic law. For example, achange in the yolx o from one to two gives ya four-fold advantage. Note also thatEq. (53) denotes the relation between thepersonnel numbers of the opposing sides butdoes not depend explicitly on time. To derive formulas that would provide us with atime dependence, we proceed as follows.We differentiate the first equation in (50)with respect to time and then use the secondequation in this system. As a result we


Z !I

Yo

o'---- ---...;;;-....-.......~~

Fig. 13

arrive at the following differential equation:dSxdt 2 - bex = o. (56)

If for the initial conditions' we take

x (0) = xo, ~ 1,=0= - byo,

then the solution to Eq. (56) can be obtained in the form

x (t) == Xo cos ~t - 1'Yo sin ~t, (57)

where ~ = Ybe and y = Vb/c.In a similar manner we can show that

y (t) = Yo cos pt - .=!. sin ~t. (58)y

Figure 13 depicts the graphs of the functions specified by Eqs. (57) and (58) in

en, 1. Construction of Differential Models 65

the case where K > 0 (i. e. by~ > cx~, orYYo > xo)·

We note in conclusion that for side y towin a victory it is not necessary that Yo begreater than X o• The only requirement isthat YYo be greater than X o•

Case B (differential systems of the B-typeand the linear law). The dynamical equations that model combat operations betweentwo opposing sides can easily be solved,as in the previous case, if we exclude thepossibility of losses not associated withcombat and if neither side receives reinforcements. Under these limitations theB-type differential system assumes the form

dx dy h(ft= - gxy, <It = - xy. (59)

Dividing the second equation in (59) bythe first, we get the equation

dy hdx=g'

which after being integrated yields

g [y (t) - Yo] == h [x (t) - xo). (60)

This linear relationship explains why thenonlinear system (59) corresponds to a model with a linear law for conducting combatoperations. Equation (60) can be rewrittenin the form

gy - hx = L,

5-0770

(61)

66 Difterential Equations in Applications

!1ft) L>0, II wills

a: wins

Fig. 14

-L/h ,x(t}

with L = gyo - ha.; This implies, for one,that if L is positive, side y wins as a resultof combat operations, while if L is negative, side x wins.

Figure 14 provides a geometrical interpretation of the linear law (61) for differentvalues of L.

Let us now study in greater detail thesituation where one of the sides wins. Weassume that this is side y. Then, as we already know, gyo - hxo must be positive, or

..!!.L > !!:- •Xo g

If we now turn to (49), we see that side ywins if

(62)

Chi i. Construction of Differential Models 67

Thus, the strategy of y is to make the ratioYo/xo as large as possible and the ratioA xlA y as small as possible. From the practical standpoint it is more convenient towrite inequality (62) in the form

Ayyo > rxArxAxxo r'IAnl·

We see that in a certain sense the productsAyYo and Axxo constitute critical values.

We note, finally, that by combining(61) and (59) we can easily derive formulasthat give the time dependence of changesin the personnel numbers of both sides.

Case C (differential systems of the C-typeand the parabolic law). In the C-model theguerilla forces face regular troops. We introduce the simplifying assumptions thatneither is supplied with reinforcements andthat the losses associated directly with combat operations are nil. In this case we havethe following system of differential equations:dxdt=-gxy,

dydt=-cx, (63)

where x (t) is the number of personnel inthe guerilla forces, and y (t) the number ofpersonnel in the regular troops. Dividingthe second equation in (63) by the first, wearrive at the differential equation

dy _ cdx -gy.

G8 Differential Equations in Applications

!I(l) M>O, II wins

-M/2c

Fig. 15

11=0, even

:crt)

Integrating with the appropriate limitsyields the following relationship:

gy2 (t) =-= 2cx (t) +M, (64)

where M == gy~ - 2cxo. Thus, the systemof differential equations (63) corresponds toa model with a parabolic law for conductingcombat operations. The guerilla forces winif M is negative; they are defeated if M ispositive.

Figure if) depicts the parabolas definedvia Eq. (64) for different values of ill.Experience shows that regular troops candefeat guerilla forces only if the ratio Yo/xoconsiderably exceeds unity. Basing our reasoning on the parabolic law for conductingcombat operations and assuming M positive, we conclude that the victory of regular


troops is guaranteed if (Yo/xo)2 is greaterthan (2c/g) x;t. If we allow for (48) and(49), we can rewrite this condition in theform

1.12 Why Are Pendulum ClocksInaccurate?

To answer this question, let us consder anidealized model of a pendulum clock consisting of a rod of length l and a load of massm attached to the lower end of the rod(the rod's mass is assumed so small that itcan be ignored in comparison to m) (Figure 16). If the load is:deflected by an angle a,tL:--- _ ... 5···;................

Ir,IIIJ _--~--s

Fig. 16


and then released, in accordance withenergy conservation we obtain

mv 2-2-=mg(lcosO-lcoset), (65)

where v is the speed at which the loadmoves, and g is the acceleration of gravity.

Since s = 18, we have v = ds/dt =l (deldt) and (65) leads us to the fo llowingdifferential equation:

l (de)22 dt =g (cos 8- COSet). (66)

If we now allow for the fact that B decreaseswith the passage of time t (for small t's),we can rewrite Eq.. (66) in ihe form .

dt= _ .. ;- l deV 2g ycos9-cosa

If we denote the period of the pendulumby T, we have

y-_O~ = - ;g ~ Ycos:~cosa '

a

or_ a.

T = 4 .. / L. \ de (67)V 2g J y cos e- cos a ·

o

The last formula shows that the period ofthe pendulum depends on a. This is the

Ch. 1. Censtruetion of Differential Models 71

main reason why pendulum clocks are inaccurate since, practically speaking, everytime that the load is deflected to the extreme position the deflection angle differsfrom a.

We note that formula (67) can be writtenin a simpler form. Indeed, since

cos e= 1-2 sin2 ~ , cos cx, = 1-2 sin'' ~ ,

we have_ a

T = 2 .. /" L \ deV g J .. / ct 8

o V sin l T-sin' 2_tt

=2V 1- ~ dO (68)g 0 V k2 - sin2 ~

with k = sin (a/2).Now instead of variable e we introduce

a new variable <p via the formulasin (8/2) = k sin cpo This implies that whene increases from 0 to a, the variable <pgrows from 0 to n/2, with

1. 62 cos 2 de= k cos


The last relationship enables us to rewriteformula (68) in the form

_1t/2

T=4 l / - l r dcp. g ~ -V1- k 2 sin 2 cp

o

=4 V : F (k, 1£/2),

where the function

,p\ dcp

F(k, '1\')= ~ V1-k2sin2qJo

is known as the elliptic integral of the firstkind, differing from the elliptic integral ofthe second kind

'tV

E(k,'I\')= ~ V1-k2 sin2 q> d q>.o

Elliptic. integrals cannot be expressed interms of elementary functions. Therefore,all further discussion of the pendulum problem will be related to an approach considered when conservative systems are studiedin mechanics. Here we only note that thestarting point in our studies will be thedifferential equation

d2S. 1/---dt2 +k SIll 8= 0, k = V gil,

which can be obtained from Eq. (60) bydifferentiating with respect to t.


1.13 The Cycloidal Clock

We have established that clocks with ordinary (circular) pendulums are inaccurate.Is there any pendulum whose period is independent of the swing? This problem wasfirst form.ulated and solved as early as the17tll century. Below we give its solution,but first let \18 turn to the derivation of theequation of a remarkable curve, whichGalilee Galilei called the cycloid (from theGreek word for "circular"). This is a planecurve generated by a point on the circumference of a circle (called the generatingcircle) as it rolls along a straight line without slippage.

Suppose that the x axis is the straightline along which the generating circle rollsand that the radius of this circle is r (Figure 17). Let us also assume that initiallythe point that traces the cycloid is at theorigin and that after the circle turns throughan angle 8, the point occupies position M.Then, on the basis of geometrical reasoningwe obtain

x = OS == OP - SP,

y = NIS = CP - CN.

But

OP = MP = r8, SP = MN = r sin 8,

CP = r, CN = r cos 8.


II

Fig. 17

Hence, parametrically the cycloid is specifled by the following equations:

x = r (8 - sin 8), y = r (1 - cos 8). (69)

If we exclude parameter 8 from these equations, we arrive at the following equationof a cycloid in a rectangular Cartesian coordinate system Oxy:

x = r cos' ( r;. Y ) - V2ry _ y2•

The very method of constructing a cycloid implies that the cycloid consists of congruent arcs each of which corresponds to afull revolution of the generating circle. *

• The reader may find many interesting facts aboutthe cycloid and related curves in G.N. Berman'shook The Cycloid (Moscow: Nauka, 1980) (inRussian).


The separate arcs are linked at 'pointswhere they have a common vertical tangent. These points, known as cusps, correspond to the lowest possible positions occupied by the point on the generating circlethat describes the cycloid. The highestpossible positions occupied by the samepoint lie exactly in the middle of each arcand are known as the vertices. The distancealong the straight line between 'successivecusps is 2nr, and the segment of thatstraight line between successive cusps isknown as the base of an arc of the cycloid.

The cycloid possesses the following properties:

(a) the area bounded by an arc of a cycloidand the respective base is thrice the area ofthe generating circle (Galileo's theorem);

(b) the length of one cycloid arc is fourtimes the length of the diameter of the generating circle (Wren's theorem).

The last result is quite unexpected, sinceto calculate the length of such a simplecurve as the circumference of a circle it isnecessary to introduce the irrational numberJ[, whose calculation is not very simple,while the length of an arc of a cycloid isexpressed as an integral multiple of the radius (or diameter) of the generating circle.The cycloid has many other interestingproperties, which have proved to be extremely important for physics and engineering.For example, the profile of pinion teeth


Fig. 18

and of many types of eccentrics, earns, andother mechanical parts of devices have theshape of a cycloid.

Let us now turn to the problem tha tenabled the Dutch physical scientist, astronomer, and mathematician Christian Huygens (1629-1695) to build an accurate clockin 1673. The problem consists of buildingin the vertical plane a curve for which thetime of descent of a heavy particle slidingwithout friction to a fixed horizontal lineis the same wherever the particle starts onthe curve (it is assumed that initially, attime t = to, the particle is at rest). Huygens found that the cycloid possesses suchan isochronous (from the Greek words for"equal" and "time") or tautochronousproperty (from tautos for "identical").

From the practical standpoint the problem can be solved in the following manner.Let us assume that a trough in the form of acycloid is cut out of a piece of wood asshown in Figure 18. "Ve take a small metalball and send it rolling down the slope.Ignoring friction, let us try to determine

en. 1. Construction of Differential Models 77

the time it takes the hall to reach the lowest possible point, K, starting from pointM, say.

Let Xo and Yo be the coordinates of theinitial posi tion of the ball, i. e. point M,and eo the corresponding value of parameter 8. When the ball reaches a pointN (8),the distance of descent along the verticalwill he h, which in view of Eq. (69) can befound in the following manner:

h = y - Yo = r (1 - cos 8)

- r (1 - cos 80) = r (cos 80 - cos 8).

We know that the speed of a falling objectis given by the formula

v= V2gh,

where g is the acceleration of gravity. Inour case the last formula assumes the form

v = V2gr (cos eo - cos 8)

On the other hand, since speed is the derivative of distance s with respect to time T,we arrive at the formula

:; = 112gr (cos 00 - cos 0)

Since for a cycloid ds == 2r sin (8/2) de, wecan rewrite this formula (which is actuallya differential equation) in the form

dT = 2r sin (8/2) de •-V 2gr (cos eo - cos 8)

78 Diilerential Equations in Applications

Integrating this equation within appropriate limits, we obtain

2r sin (8/2) de

°J/2gr (cos eo - cos 0)

_ n

== _ 2 .. / I: \ ._ d cos (0/2)V g J . e e

o 1./ cos2 _0-cos2-~ 2 2

/-=:rt l rig.

Thus, the time interval T in the courseof which the ball rolls down from point Mto point K is given by the formula

T = nVrlg,

which shows that period T is independentof 80' that is, the period does not dependon the initial position of the ball, M.Obviously, two balls that begin their motion simultaneously from points M and Nwill roll down and find themselves at pointK at the same moment of time.

Since we agreed to ignore friction, in itsmotion down the slope the ball passes pointK and, by inertia, continues its motionup the slope to point M 1 lying at the samelevel as point AI. Proceeding then in theopposite direction and traversing its path,the ball completes a full cycle. This constitutes the motion of a cycloidal pendulum

ChI 1. Const.ruction of Differential Models 79

Fig. 19

with a period of osci lla lions

To = 4nVrlg. (70)

A peculiar feature of the cycloidal pendulum, which sets it apart from the simple(circular) pendulum, is that its period doesnot depend on the amplitude (or swing).

Let us now show how an ordinary circular pendulum can be made to move in atautochronous manner without resorting totrough and similar devices with considerab Ie friction. To this end it is sufficientto make a template (say, out of wood) consisting of two semiarcs of a cycloid with acommon cusp (Figure 19). The templateis fixed in the vertical plane and a stringwith a ball is suspended from the cusp O.The length of the string must be twice thediameter of the generating circle of thecycloid.

If the ball on the string is deflected to anarbitrary point M, it begins to swing backand forth with a period that is independ-

80 Differential Eqnations in Applications

Fig. 20

A

c

ent of the positionof point M. Evenif due to frictionand air drag theamplitude of theoscillations diminishes, the periodremains unchanged. For a circu

B lar pendulum,E which moves along

an arc of a cir-curnference, the isochronous proper-

ty is satisfied approximately only forsmall amplitudes, when the arc differslittle from the arc of a cycloid.

By way of an example let us study thesmall oscillations that a pendulum executes

along the arc .iB of a cycloid (Figure 20).If these oscillations are very small, theeffect of the guiding template is practicallynil and the pendulum oscillates almost likean ordinary circular pendulum with a string(the "rod") whose length is 4r. The path ABof a cycloidal pendulum will differ littlefrom the path CE of a circular pendulumwith a string length equal to 4r. Hence, theperiod of small oscillations of a circularpendulum with a string length 1 = 4r ispractically the same as that of a cycloidalpendulum of the same length.


Now, if in (70) we put r = l/4, the periodof small oscillations of a circular pendulum can be expressed in terms of the lengthof the string:

T = 2n VZ/g.This formula is derived (in a different way)in high school physics.

In conclusion we note that the cycloidis the only curve for which a particle moving along it performs isochronous oscillations.

1.14 The Bracbistochrone Problem

The problem concerning the brachistochrone(from the Greek words for "shortest" and"time"), a curve of fastest descent, was proposed by the Swiss mathematician JohnBernoulli (1667-1748) in 1696 as a challengeto mathematicians and consists of the following.

Take two points A and B lying in a vertical plane but not on a single vertical line(Figure 21). Among the various curves passing through these two points we must find

~ B

Fig. 21

6-0770


Fig. 22

the one for which the time required for aparticle to fall from point A to point Balong the curve under the force of gravity isminimal.

The problem was tackled by the bestmathematicians. It was solved by JohnBernoulli himself and also by Gottfried W.von Leibniz (1646-1716), Sir Isaac Newton(1642-1727), Guillaume L'Hospital (16611704), and Jakob Bernoulli (1654-1705).The problem is famous not only from thegeneral scientific viewpoint but also forbeing the source of ideas in a completelynew field of mathematics, the calculus ofvariations.

Solution of the brachistochrone problemcan be linked with that of another problemoriginating in optics. Let us turn to Figure 22, in which a ray of light is depicted 8S

propagating from point A to point P with

Ch. [. Construction of Differential MOdels 83

a velocity V1 and then from point P topoint B in a denser medium with a lower velocity v 2 • The total time T required for theray to propagate from point A to point Bcan, obviously, be found from the followingformula:

Va2+x' Yb2+ (c -':'X)2T= + .

VI v2

If we assume that the ray of light propagates from point A to point B along this pathin the shortest possible time T, the derivative dT/dx must vanish. But then

x c-x

VI ]Ia2+x2

orsin al sin CX2

VI Vs

The last formula expresses Snell's wellknown law of refraction, which initiallywas discovered in experiments in the formsin <:Xl/sin a 2 = a, with a constant.

The above assumption that light choosesa path from A to B that would take theshortest possible time to travel is known asFermat's principle, or the principle of leasttime. The importance of this principle liesnot only in the fact that it can be taken asa rational basis for deriving Snell's law butalso, for one, in that it can be applied tofinding the path of a ray of light in a medium of variable density, where generally


Fig. 23

the light travels not along straight-linesegments.

For the sake of an example let us consider Figure 23, which depicts a ray of lightpropagating through a layered medium. Ineach layer the speed of light is constant,but it decreases when we pass from an upper layer to a lower layer. The incident rayis refracted more and more strongly as itpasses from layer to layer and moves closerand closer to the vertical line. ApplyingSnell's law to the interfaces between thelayers, we get

Now let us assume that the layer thicknesses decrease without limit while the number of layers increases without limit. Then,in the limit, the velocity of light changes(decreases) continuously and we conclude


Fig. 24

(see Figure 24) that

sin ee-v-==a, (71)

with a == const. A similar situation is observed (with certain reservations) when aray of SUD light falls on Earth. As the raytravels through Earth's atmosphere of increasing density, its velocity decreases andthe ray bends.

Let us go back to the brachistochroneproblem. We introduce a system of coordinates in the vertical plane in the wayshown in Figure 25. We imagine that aball (like a ray of light propagating in media) is capable of choosing the path of descent from pointA to pointB with the shortest possible time of descent, Then, as [ol-


Ak------r-----~

Fig. 25

lows from the above reasoning, formula (71)is valid.

The energy conservation principle impliesthat the speed gained by the ball at agiven level depends only on the loss of potential energy as the ball reaches the leveland not on the shape of the trajectory fol-lowed. This means that v = y2gy.

On the other hand, geometric construction enables us to show that

· A 1 1SIn ex == cos P = --A-= --;:==~

seep V1+tan2p

1- 1/1+ (y')2 •

Combining the last two relationships with(71) yields

y (1+ (11')2] == C, (72)

This constitutes the brachistochrone equation. We wish to demonstrate that only acycloid can represent a brachistochrone.Indeed, since y' = dy/dx, by dividing thevariables in Eq. (72) we arrive at the equation

dx = ( C y y ) 1/2 dy.

We introduce a new variable cp via thefollowing relationship:

(Y ) 1/2

C-y = tan cp.

Then

y = Csin2 cp, dy = 2Csin drp,

dx = tan fP dy = 2C sin2 cp dcp= C (1- cos 2q» dcp.

Integration of the last equation yields

x = ; (2q> - sin 2q» +Ct ,

where, in view of the initial conditions,z = y = 0 at cp = 0 and C1 = O. Thus,

x = ; (2q>- sin 2q»,

y=Csin2q>= ~ (1-cos2q».

Assuming that C/2 = rand 2(p = 8, wearrive at the standard parametric equations


of the cycloid (69). The cycloid is, indeed, aremarkable curve: it is not only isochronous-it is brachistochronous.

1.15 The Arithmetic Mean, theGeometric Mean, and theAssociated Differential Equation

Let us consider the following curious problem first suggested by the famous Germanmathematician Carl F. Gauss (1777-1855).

Let mo and no be two arbitrary positivenumbers (mo > no)' Out of mo and no weconstruct two new numbers ni, and n1 thatare, respectively, the arithmetic mean andthe geometric mean of m., and no. In otherwords, we put

mo+no ... /--mt = 2 ' n t = V mono'

Treating ml and n1 in the same manner asm ; and no, we put

ml+nl 1/--m2 == 2 ' n2 = Y mint·

Continuing this process indefinitely, we arrive at two sequences of real numbers, {mk}and {nk} (k = 0, 1, 2, .), which, as caneasily be demonstrated, are convergent. Wewish to know the difference of the limits ofthese two sequences.

An elegant solution of this problemamounting to setting up a second-order lineardifferential equation is given below. 1t be-


longs to the German mathematician CarlW. Borchardt (1817-1880). Let a be thesought difference. I t obviously depends onmo and no, a fact expressed analytically asfollows: a == f (mo, no), where t is a function. The definition of a also implies thata == f (m 1 , n l ) . Now, if we multiply moand no by the same number k, each of thenumbers ms, n l , m«. 112 , introducedabove, including a, will be multiplied byk. This means that a is a first-order homogeneous function in m.; and no and, hence,

a == mot (1, nolm o) = mit (1, n1lm}).

Introducing the notation x = no/m o, Xl =n1/mt, *, Y == 1lt (1, nolmo), Yl =1/t (1, nt/rnl) , we find that

mo 2Yl 3Y == Yt ml == 1+x · (7 )

Since Xl is related to x via the equation

2 yxXi = 1+x 'we find that

dXl f -x (xl- x~) (1 +x)2(i"X== (1+x)2 II x == 2 (.1; -- x3)

On the other hand, Eq. (73) leads to thefollowing rel ationship:

dy 2 + 2 dYl dXl

d~ == - (1+x)2 Yl 1+x dXl dX'

Replacing dx1/dx with its value given bythe previous relationship and factoring out(x - x3

) , we find that

( S) dy_2x(x-1)x-x dx- 1+x Yl

+(1+X)(Xl-X~) ddYl

•Xl

Differentiating both sides of this equationwith respect to x, we find that

d [<X-X3)-*"]dx

[x(x-1) ]

-2 d 1+x +2x(x-1) dYl dXl- Yl dx 1+x dXl dx

+ (x t - ~~~) ddYIXl

+(1 + x) d:1[(x 1 - x~) ~~: ] ~:l .By elementary transformation this equation can be reduced to

ddx [(X-X3) ~~J -xy

1- x {d [( 3) dYl ] }= (1+ x) yx dXl X t -- Xl dXl - XtYt •

If we now replace x with Xl' then Xl willbecome x 2• If we then replace Xl with x 2,

cu. 1. Construction of Differential Models 91

we find that X 2 is transformed into x 3' etc.Hence, assuming that

d: [(x- x3) :~J - xy = a* (y),

we arrive at the following formula:

a*(y)= i-x i-x i-x2

(i+x) Vx (1+xl) VXl (1+x2) YX2

1-xn *( )X X (1-{-xn) VX n a Yn·

If we now send n to infinity, we find that1 - X n tends to zero and, hence,

a* (y) = o.This means that y satisfies the differentialequation

d2y dy(x - x3

) dx 2 +(1 - 3x2) dx - xy = O. (74)

If we note that

we can easily find the value of this number. Indeed, since y must be the constantsolution to Eq. (74), we find that onlyy == 0 is such a solution. Thus, the difference of the limits of the sequences {mIt} and{nit} is zero.

The differential equation (74) is remarkable not only because it enabled reducing

the initial problem to an obvious one butalso because it is linked directly to the solution of the problem of calculating the period of small oscillations of a circular pendulum.

As demonstrated earlier, the period ofsmall oscillations of a circular pendulumcan be found from the formula

T = 4 Vl/g F (k, n/2),

with

n/2

F (k, n/2) = ~o

dcp

It has been found that if 0 ~ k < 1, then

n/2

~o

dq>V1-k2 sin2 rp

where

n=1

is a solution to the differential equation (74).

ct. 1. Construction 01 Differential Models 93

1.16 On the Flight of an ObjectThrown at an Angle to the Horizon

Suppose that an object is thrown at an angle ex, to the horizon with an initial velocityDo- We wish to derive the equation of theobject's motion that ignores forces offriction (air drag).

Let us select the coordinate axes as shownin Figure 26. At an arbitrary point of thetrajectory only the force of gravity Pequal to mg, with m the mass of the objectand g the acceleration of gravity, acts onthe object. Hence, in accordance withNewton's second law, we can write thedifferential equations of the motion of theobject as projected on the x and y axes asfollows:

d2ymdt2=-mg.

Factoring out m yields

d1x d2ydt2=O, dt2== -g. (75)

m•~mg

Fig. 26

(76)

94 Differentie] Equations in Applications

The initial conditions imposed on the object's motion are

dxx=O, y=O, (it==vocosa,

dy • t 0(it=voslna a t= .

Integrating Eqs. (75) and allowing for theinitial conditions (76), we find that theequations of the object's motion are

x = (vo cos ex) t, Y = (vo sin ex) t - gt 2/2.(77)

A number of conclusions concerning thecharacter of the object's motion can bedrawn from Eqs. (77). For example, we canfind the time of the object's flight up to themoment when the object hits Earth, the rangeof the flight, the maximum height that theobject reaches in its flight, and the shapeof the trajectory.

The first problem can be solved by findingthe value of time t at which y = O. Thesecond equation in (77) implies that thishappens when

t [ Vosin ex - g~ ] = 0,

that is, either t = 0 or t = (2vo sin rx)/g.The second value provides the solution.

The second problem can be solved bycalculating the value of x at a value of tequal to the time of flight. The first equa-


tion in (77) implies that the range of theflight is given by the formula

(vocos a) (2vo sin a)g

v~ sin 2ag

This implies, for one thing, that the rangeis greatest when 2a == 90°, or a == 45° Inthis case the range is v:1g.

The solution to the third problem can beobtained immediately by formulating themaximum condition for y. But this meansthat at the point where y is maximal thederivative dyldt vanishes. Noting that

dy t+.dt== - g VoSID~,

we arrive at the equation -gt + Vosin a =0,which yields

t = (vo sin a)/y.

Substituting this value of t into the secondequation in (77), we find that the maximumheight reached by the object is v~ sin" a/2g.

The solution to the fourth problem hasalready been found. Namely, the trajectoryis represented by a parabola since Eqs. (77)represent parametrically a parabola, whichin rectangular Cartesian coordinates canbe written as follows:

gz2y=xtana- -22 sec2a.

Vo

98 OiHerential Equations in Applications

1.17 Weightlessness

The state of weightlessness (zero g) can beachieved in various ways, although it isassociated (consciously or subconsciously)with the "floating" of astronauts in thecabin of a spacecraft.

Let us consider the following problem.Suppose that a person of weight P is standing in an elevator that is moving downward with an acceleration (J) = txg; witho< ex, < 1 and g the acceleration of gra vity. Let us determine the pressure thatthe person exerts on the cabin's floor andthe acceleration that the elevator mustundergo so that this pressure will vanish.

Two forces act on the person in the elevator (Figure 27): the force of gravity P andthe force Q that the floor exerts on the per-

a.

p

X

Fig. 27

son (equal numerically to the force of pressure of the person on the floor). The differential equation of the person's motion canbe written in the form

d2xmCfj2=P- Q. (78)

Since d2xldt2 = 0) = ag and m = Pig, wecan rewrite Eq. (78) as follows:

d2xQ==P-mCfj2==P(1-a,). (79)

Since 0 < a, < 1, we conclude that Q< P.Thus, the pressure that the person exerts onthe floor of the cabin of an elevator moving downward is determined by the force

Q = P (1 - a,).

On the other hand, when the elevator is moving upward with an acceleration 0) = ag;o< a < 1, the pressure that the personexerts on the floor of the cabin is determined by the force Q = P (1 + a). Let usnow establish at what acceleration the pressure vanishes. For this it is sufficient to putQ = 0 in (79). We conclude that in thiscase a, = 1, that is, for Q to vanish theacceleration of the elevator must be equalto the acceleration of gravity.

Thus, when the cabin is falling freelywith an acceleration equal to g, the pressure that the person exerts on the floor isnil. It is this state that is called uieightless-7 -0770


ness. In it the various parts of a person'sbody exert no pressure on each other, sothat the person experiences extraordinarysensations. In the state of weightlessnessall points of an object experience the sameacceleration.

Of course, weightlessness is experiencednot only during a free fall in an elevator.For illustration let us consider the followingproblem.

What must be the speed of a spacecraftmoving around the Earth as an artificialsatellite for a person inside it to be in thestate of weightlessness?

One assumption in this problem is thatthe spacecraft follows a circular orbit ofradius r + h, where r is Earth's radius, andh is the altitude at which the spacecrafttravels (reckoned from Earth's surface). Theprevious problem implies that in the stateof weightlessness the pressure on the wallsof the spacecraft is zero, so that the force Qacting on an object inside the spacecraftis zero, too. Hence, Q = O. Let us nowturn our attention to Figure 28. The x axisis directed along the principal normal nto the circular trajectory of the spacecraft.We use the differential equation of the motion of a particle as projected on the principal normal:


m

Fig. 28

n

where p == r + h, Lj F h n = F, and F ish=1

directed along the principal normal to thetrajectory. VVe get

~-F-- m d2x

r+h - -" dt 2 '

or the eqnation

d2x v2

dj2- r+h

Substituting this value of d2x/dt2 intoEq. (78), we find thatmv2

r+h ==P-Q. (80)


Here force P is equal to the force F ofattraction to Earth, which, in accordancewith Newton's Iaw of gravitation, is inversely proportional to the square of the distance r -~ h from the center of Earth,that is,

kmF == (r+h)2 ,

where m is the mass of the spacecraft, andconstant k can be determined from thefollowing considerations. At Earth's surface, where h = 0, the force of gravity Fis equal to mg. The above formula thenyields k === gr": Hence,

mgr2

P = F= (r+h)2 ,

where g is the acceleration of gravity atEarth's surface.

If we now substitute the obtained valueof Pinto (80) and note that Q === 0, we findthat the required speed is given by theformula

v= r'V r~h

1.18 Kepler's Laws of PlanetaryMotion

In accordance with Newton's law of gravitation, any two objects separated by a distance r and having masses m and Mare at-


y

~- m

/'

f/./' I

./' IY/' I

M ./ I!fl

0 X P X

Fig. 29

tracted with a force

F = G;~M , (81)

where G is the gravitational constant.Basing ourselves on this law, let us de

scribe the motion of the planets in the solarsystem assuming that m is the mass of aplanet orbiting the Sun and M is theSun's mass. The effect of other planets onthis motion will be ignored.

Let the Sun be at the origin of the coordinate system depicted in Figure 29 andthe planet be at time t at a point withrunning coordinates x and y. The attractiveforce F acting on the planet can be decomposed into two components: one parallel tothe x axis and equal to F cos q> and theother parallel to the y axis and equal to

F sin rp, Using formula (81) and Newton'ssecond law, we arrive at the followingequations:

•• GmMmx === - F cos (p == - -r-2 - cos tp, (82)

•• • Gl1~M •my == - F SIll rp == - -r2- SIn cp. (83)

Bearing in Blind that sin === x/r, we can rewrite Eqs. (82) and(83) as follows:

kxx=-r 3 ,

where constant k is equal to GM.Finally, allowing for the fact that r ===

y x2 + y2, we arrive at the differentialequations

kyy == - (x2+ y2)3/2 •

(84)

kx

Without loss of generality we can assumethat

. .x === a, y === 0, x === 0, y === Do at t === O. (85)

We see that the problem has been reduced to the study of Eq. (84) under theinitial conditions (85). The special features of Eqs. (84) suggest that the most convenient coordinate system here is the oneusing polar coordinates: x = r cos rp and

y == r sin cp. Then· . .x == r cos cp - (r sin (p) cp,· . .y == r sin cp + (r cos <p) qi,.. .. . .x == r cos rp - 2 (r sin ({)) rp

- (r sin qJ) ·(P - (r cos qJ) ~2,·. .. ..y == r sin qJ + 2 (r cos qJ) q>

.. .+ (r cos cp) {p - (r sin q» q>2.

Hence,

(86)

·. ... ..x == (r - rc(2) cos cp - (2r cp + r cp) sin cp,

if· == (~. - r~2) sin cp + (2; ~ + ~~) cos cp.

Using the last two relationships, we canrewrite the differential equations (84) inthe form

.. . ..(r - r«2) cos cp - (2r<p + r rp) sin <p

k cos fPr2

.. .(r - r<p2) sin {p + (2r<p+ r q» cos <p

k sin (pr 2

(87)

(88)

Multiplying both sides of Eq. (87) bycos fj), both sides of Eq. (88) by sin cp, and

adding the products, we find that

; •- r~2 = -klr2• (89)

Multiplying both sides of Eq. (87) bysin <p, both sides of Eq. (88) by cos <p, andsubtracting the second product from thefirst, we arrive at the equation..2rcp + rep == O. (90)

As for the initial conditions (85), in polarcoordinates they assume the form

r = a, <p == 0, ; == 0, ~ == vola at t = O.(91)

Thus, we have reduced the problem ofstudying Eqs. (84) under the initial conditions (85) to that of studying Eqs. (89) and(90) under the initial conditions (91). Wealso note that Eq. (90) can be rewritten inthe form

d •dt (r2cp) = O. (92)

But Eq. (92) yields

r 2cp == C1 , (93)

where C1 is a constant possessing an interesting geometric interpretation. Precisely, suppose that an object moves from

point P to point Q along the arc PO (Figure 30). Let S be the area of the sector limited by the segments OP and OQ and the

!Ja

o---.....-----.........-~P &C

Fig. 30

arc PQ. Frorn the calculus course we }\:IlOW

that

or dS == (1/2) r2 dip. Hence,

dS 1 d~ 1 •(It==Tr2(It=Tr2<P. (94)

The derivative dS/dt constitutes what isknown as the areal velocity, and since in

view of (93) r2cp is a constant, we concludethat the areal velocity is a constant, too.But this, in turn, means that the objectmoves in such a manner that the radiusvector describes equal areas in equal timeintervals. This law 0/ areas constitutes oneof the three Kepler laws. In full it can beformulated as follows: each planet movesalong a plane curve around the Sun in sucha manner that the radius rector connecting


the Sun with the planet describes equal areasin equal time intervals.

To derive the next Kepler law, whichdeals with the shape of the planetary trajectories, we return to Eqs. (89) and (gO)with the initial conditions (91) imposed onthem. The initial conditions imply, for

one, that r == a and ~ == vola at t == O.But then condition (93) implies that C1 ==avo. Hence,

r2~ = avo, or ;p == avolr2. (D5)

This transforms Eq. (89) intoa2v~ k

r == --,:3 -T2.

Assuming that r == p, we can rewrite thisequation in the form

~_~~~ ~_a2v~_~dt - dr dt - P dr - r3 r 2 ,

or

iE..- a2v~ _ ~P dr - r3 r2

Separating the variables in the last differential equation and integrating, we arriveat the following relationship:p2 k a2v2

2==-;:- ~ 2r2o +Cz •

Since p == r == 0 at r == a, we find that

C _ v~ k2-2-a-·

(96)

Ch. 1. Construction of Differential l\1odeIs 107

Thus, we arrive at the equation

r'l. k a2v2 v2 k_ u+ 02-7- 2r2 2---0:'

Of, if we consider uuly the positive valueof the square root,

~== -. I(v _ 2k )+~ __ a2v~dt V 0 a r r2

Dividing Eq. (96) by Eq. (95), we find that

~ ==r ¥ar2 -4- 2Ar - 1d~ I P ,

where1 2k k

(J., ::= ll'2 - a3v2 ' ~ == a2v2o 0

The last equation can be integrated by substituting flu for r. The result is

r-- a2v~/k- 1+ecos(cp+Cs) ,

where e = ·Va + ~2/~ = av~/k - 1. Theconstant C3 can be determined from thecondition that r = a at rp = O. It is easyto verify that Ca = O. Thus, we finally have

a2v2 /kr = 1+e ~os (P (97)

From analytic geometry we know that thisis the equation of a conic in terms of polarcoordinates, with e the eccentricity of theconic. The following cases are possible

here:(1) an ellipse if e<1, or v~<2k/a,

(2) a hyperbola if e > 1, or v~ > 2kla,(3) a parabola if e = 1, or v~ = Zkla,(4) a circle if e = 0, or v~ = lela.

Astronomical observations have shownthat for all the planets belonging to thesolar system the value of v~ is smallerhan 2kla. We, therefore, arrive at anotherof Kepler's laws: the planets describe ellipses with the Sun at one focus.

Note that the orbits of the Moon andthe artificial satellites of Earth are alsoellipses, but in the majority of cases theseellipses are close to circles, that is, e differslittle from zero.

As for recurrent comets, like, say, Halley's comet, their orbits resemble "prolate"ellipses whose eccentricity is smaller thanunity but very close to it. Say, Halley'scomet appears in Earth's neighborhoodapproximately every 76 years. Its latestapparition was in the period between theend of 1985 and the beginning of 1986.

Celestial bodies that move along parabolic and hyperbolic orbits may be observedonly once, since they never return tothe same place.

Let us now establish the physical meaning of eccentricity e. First we note that

the components ~ and if of a planet's velocity vector Valong the x and y axes,.respec-

Ch. 1. Construction of Differential l\fodels 109

tively, and the size v of vector V satisfythe relationship

. .v2 === x2 + y2,

which when combined with (86) yields

v2 == r2cp2 + r2

•

From this it follows that the kinetic energyof a planet of mass m is given by the formula1 1 .•

"2 mv 2 ="2 m (r2<p2 + r 2) . (98)

The potential energy of a system is minusone times the amount of work needed tomove the planet to infinity (where the potential energy is zero by convention).Hence,

00

_ \' km dr == km Joo == _ km~ r2 r r rr

(99)

If by E we denote the total energy of thesystem, which in view of energy conservation must be constant, then formulas (98)and (99) yield1 • • kmT m [r 2<p2+ r 2) --r-==E. (100)

Assuming that cp == 0 and combining (97)with (100), we get

a2v2o/k mr2a2v2 kmr:..::::-- 0 -E1+e' 2r4 - -r-- .

Excluding r from the last two relationships,we find that

e == .. / 1+E 2a2v5V lnk 2

Equation (97) for the shape of the orbitfinally assumes the form

a2v~/kr = ---"7==========-

1+V1+ E (2a2v~/mk2) cos cp

This formula implies that the orbit is anellipse, hyperboba, parabola, or circle if,respectively, E < 0, E > 0, E == 0, orE = -mk2/2a2v~. Thus, the shape of aplanet's orbit is completely determined bythe value of E. Say, if we could impartsuch a "blow" to Earth that it would increaseEarth's total energy to a positive value,Earth would go over to a hyperbolic orbitand leave the solar system forever.

Let us now turn to Kepler's third law.This law deals with the period of revolution of planets around the Sun. Takinginto account the results obtained in deriving the previous Kepler law, we naturallyrestrict our discussion to the case of elliptic orbits, whose equation in terms of Cartesian coordinates, as is known, is

x2 + y2 _12 tf -1,


!I

Fig. 31

where (Figure 31) the eccentricity e = C/;with C2 == ~2 - 11 2

, so thate2 == (;2 _ 'Y}2)/~2,

or

'Y}2 == ~2 (1 - e2) . (101)

Combining this with (97) and allowing forthe properties of an ellipse, we concludethat

~~-.!- ( a2v5/k + a2v~/k ) == a2vg

2 1+e 1-e 'c(1-e2)

_ a2v~£2

- k11 2 ,

or

2 a2vij;11 =-k-· (102)

1.12 Differential Equations in Applications

Let us denote the period of revolution of aplanet by T. By definition, T is the timeit takes the planet to complete one full orbit about the Sun. Then, since the area limi ted by an ellipse is 1t ~ll, we conclude, onthe basis of (94) and (95), that rt £11 =avoT/2. Finally, taking into account (102),we arrive at the following result:

T2= 4n2s2'll2 == 4n2 S3.a2v~ k

This constitutes the analytical descriptionof Kepler's third law: the squares of theperiods of revolution of the planets are proportional to the cubes of the major axes of theplanets' orbits.

1.19 Beam Deflection

Let us consider a horizontal beam AB(Figure 32) of a constant cross section andmade of a homogeneous material. Thesymmetry axis of the beam is indicated inFigure 32 by a dashed line. Suppose thatforces acting on the beam in the verticalplane containing the symmetry axis bendthe beam as shown in Figure 33. These forcesmay be the weight of the beam itself oran external force or the two forces actingsimultaneously. Clearly, the symmetry axiswill also bend due to the action of theseforces. Usually a bent symmetry axis iscalled the elastic line. The problem of deter-


mining the shape of this line plays an important role in elasticity theory.

Note that there can be various types ofbeam depending on the way in which beamsare fixed or supported. For example, Figure 34 depicts a beam whose end A is rigidlyfixed and end B is free. Such a beam is saidto be a cantilever. Figure 35 depicts a beamlying freely on supports A and B. Anothertype of beam with supports is shown inFigure 36. There are also various ways inwhich loads can be applied to beams. Forexample, a uniformly distributed load isshown in Figure 34. Of course, the load canvary along the entire length of the beamor only a part of this length (Figure 35).Figure 36 illustrates the case of a concentrated load.

Let us consider a horizontal beam OA(Figure 37). Suppose that its symmetryaxis (the dashed line in Figure 37) lies onthe x axis, with the positive directionbeing to the right of point 0, the origin.The positive direction of the y axis isdownward from point O. External forcesF 1 , F 2 , (and the weight of the beamif this is great) bend the symmetry axis,which becomes the elastic line (also depicted in Figure 38 by a curved dashed line).The deflection y of the elastic line from thex axis is known as the sag of the beam atpoint x. Thus, if we know the equation ofthe elastic line, we can always find the sag

8-0770


~ -----------~BA ~Wff/Ul//;//!M'IJ(Z///}j///)//////~

Fig. 32

A~n7T77~B

Fig. 33

1 BA~

Fig. 34

of the beam. Below we show how this canbe done in practical terms.

Let us denote by M (x) the bending moment in the cross section of the beam atcoordinate x, The bending moment is defined as the algebraic sum of the momentsof the forces that act from one side of thebeam at point x. In calculating the moments we assume that the forces acting onthe beam upward result in negative moments while those acting downward resultin positive moments.

Ch, 1. Construction of Differential Models ~15

Fig. 35

Fig. 36

Fig. 37

The strength-of-materials course provesthat the bending moment at point x is related to the curvature radius of the elasticline via the equation

y"EJ (1+(y')2]Sj2 M (x), (103)

where E is Young 's modulus, which de-8*

t 16 Differential Equations in Applications

Fig. 38

pends on the type of material of the beam,J is the moment of inertia of the cross section of the beam at point x about the horizontal straight line passing through thecenter of mass of the cross section. Theproduct EJ is commonly known as theflexural rigidity, in what follows we assumethis product constant.

Now, if we suppose that the sag of thebeam is small, which is usually the case inpractice, the slope y' of the elastic line isextremely small and, therefore, instead ofEq. (103) we can take the approximateequation

Ely" = M (x). (104)

To illustrate how Eq. (104) is used inpractice, we consider the following problem. A horizontal homogeneous steel beamof length l lies freely on two supports and


pl/2 1'1,/2

Fig. 39

A

sags under its own weight, which is p kgfper unit length. We wish to determine theequation of the elastic line and the maximal sag.

In Figure 39 the elastic line is depictedby the dashed curve. Since we are dealingwith a two-support beam, each support actson the beam with an upward reaction forceequal to half the weight of the beam (orpl/2). The bending moment M (x) is thealgebraic sum of the moments of these forcesacting on one side from point Q (Figure 39). Let us first consider the action offorces to the left of point Q. At a distance xfrom point Q a force equal to pl/2 acts onthe beam upward and generates a negativemoment. On the other hand, a force equalto px acts on the beam downward at adistance x/2 from point Q and generates apositive moment. Thus, the net bending

118 .Differential Equations in Applications

moment at point Q is given by the formula

M ( pl ( X) px2 plxx)= -Tx+px 2 ==-2---2-

(105)

If we consider the forces acting to theright of point Q, we find that a force equalto p (l - x) acts on the beam downward ata distance (l - x)/2 from point Q and generates a positive moment, while a forceequal to pl/2 acts on the beam upward at adistance l - x from point Q and generatesa negative moment. The net bending moment in this case is calculated by the formula

l-x ptM (x) =' p (l- x) -2- -"2 (l-x)

px2 plx==-2--T (106)

Formulas (105) and (106) show that thebending· moments prove to be equal. Now,knowing how to find a bending moment, wecan easily write the basic equation (104),which in our case assumes the form

EJ "_ px2 plxy----2 2(107)

Since the beam does not bend at points 0and A, we write the boundary conditionsfor Eq. (107) so as to find y:

y == 0 at x = 0 and. y = 0 at x ~ l.


Integration of Eq. (107) then yields

Y= dEl (x4-2lx3+Px). (108)

This constitutes the equation of the elasticline. It is used to calculate the maximalsag. For instance, in this example, basingour reasoning on symmetry considerations(the same can be done via direct calculations), we conclude that the maximal sagwill occur at x = l/2 and is equal to5pl4/384EJ, with E == 21 X 105 kgf/cm"and J = 3 X 104 em",

1.20 Transportation of Logs

In transporting logs to saw mills, loggingtrucks move along forest roads some of thetime. The width of the forest road is usually such that only one truck can travel alongthe road. Sections of the road are made wider so that trucks can pass each other. Ignoring the question of how traffic should bearranged that loaded and empty traffictrucks meet only at such sections, let usestablish how wide the turns in the roadmust be and what trajectory the drivermust try to follow so that, say, thirty-meter logs can be transported. It is assumedthat the truck is sufficiently maneuverableto cope with a limited section of the road.

Usually a logging truck consists of atractor unit and a trailer connected freely

x

120

Fig. 40

Differential Equations in Applications

R C Q

S~p

~S R P FF

to each other. The tractor unit has a front(driving) axle and two back axles abovewhich a round platform carrying two postsand a rotating beam are fastened. Thisplatform can rotate in the horizontal planeabout a symmetrically positioned verticalaxis. One end of each log is placed on thisplatform. The trailer has only two backaxles, but also has a platform with twoposts. The other ends of the logs are put onthis platform. The trailer's chassis consistsof two metal cylinders one of which canpartly move inside the other. The chassisconnects the back platform with the axisthat links the trailer with the tractor unit.Thus, the length of the chassis can changeduring motion, which enables the tractorunit and the trailer to move independentlyto a certain extent. Schematically the logging truck is depicted in Figure 40. Thepoints A and B correspond to the axes ofthe platforms a distance h apart. By XYwe denote a log for which AX == 'Ah. PointC corresponds to a small axis that connects


the tractor unit with the trailer, withAC == ah. Usually a = 0.3, but in thesimplest case of log transportation a = O.Next, FF is the front axle of the tractorunit, and PP and QQ are the back axles ofthe tractor unit, while RR and SS arethe trailer axles. All axles have the samelength 2£, so that for the sake of simplicity we assume that the width of the logging truck is 2L. As for the width of theload in its rear, we put it equal to 2W.In what follows we will need the conceptof the sweep of the logging truck, which iscommonly understood to be the maximumdeflection of the rear part of the loggingtruck (for the sake of simplicity we assumethis part to be point X) from the trajectoryalong which the logging truck moves. Letus suppose that the road has a width of2~h and that usually a turn in the road issimply an arc of a circle of radius hlo:centered at point 0 (Figure 41). For thesake of simplicity we assume that a loggingtruck enters a turn in the road in such away that the tractor unit and the trailerare positioned along a single straight line,the driver operates the truck in such a waythat point A, corresponding to the axis ofthe front platform, is exactly above theroad's center line. Point A in Figure 41is determined by the angle 'X that the truckAC makes with the initial direction. Hereit is convenient to fix a coordinate system


x------- .........

Initial~

direcfion

y

:c

Fig. 41

Oxy in such a way that the horizontal axispoints in the initial direction and the vertical axis is perpendicular to it. In a generalsituation the load carried by the truckwill make an angle ewith the initial direction. As for angle BAG in Figure 41,denoting it by u we find that u = 'X - 8.Usually this angle is the logging truck'sangle of lag. The required halfwidth h ofthe road, which determines the sweep of thelogging truck at a turn and is known as thehal/width of the road at the outer curb of a


turn, is defined as the algebraic sumOX - OA + W, while the halfwidth of theroad at the inner curb of a turn is definedas the algebraic sum OA + L - OP, whereOP is the perpendicular dropped frompoint 0 onto AB.

We stipulate that in its motion a loggingtruck's wheel either experiences no lateralskidding at all or the skidding is small.This requirement, for one thing, means thatthe center line A C of the tractor unit constitutes a tangent to the arc of a circle atpoint A, so that OA is perpendicular to A C,and angle 'X is determined by the motion ofpoint A along the arc of the circle. Note,further, that in building a road the curvature of a turn in the road is determined byan angle N° that corresponds to an arclength of a turn of approximately 30 m.In our notation,

NO. == 1800

30an h '

(109)

where h is measured in DIeters. Thus, ath = 9 m and a == 0.1 we have N° ~ 19°,while at h == 12 m and o: = 1.0 we haveN° ~ 142°. For practical considerations wemust consider only SUCII o: 's that lie between 0 and 1, and the greater the value ofu.., the greater the maneuverabil itv of thelogging truck.

The log length 'Ah will be greater than h,but again, reasoning on practical grounds,

it must not be greater than 3h. Thus, thevalue of A varies between 1 and 3. As forconstant a, it is assumed that 0 ~ a < 0.5.Finally, we note that in each case thevalue of h is chosen differently, but it variesbetween 9 and 12 ID.

Since the wheels of the tractor unit donot skid laterally, the coordinates of pointA in Figure 41 are

h · 'V h 'Vx= -SIll,."" y=-COSA.ex, ex,

The coordinates of point Bare

X=~sinX-hcose,ex,

y::; ~cosX +hsin6. (110)ex,

Since the trailer's wheels do not skid either,point B moves in the direction Be and

dYdX = -tan ,!" (111)

where 'l' is the angle which Be makes withthe initial direction. Next, employing thefact that 'X = u + e and studying triangleABC, we arrive at the following chain ofequalities:

sin (X-'I') _ sin (8-'1') _ sin u (112)h ah~'

where 0 < b < 1, and '1', 6, and u are


functions of X. If we combine (111) with(110), we find that

( - ~ sin X+ h :~ cos fl ) cos 1jJ

+( ~ cos X+ h :~ sin fl) sin 1J' = O.

Carrying out the necessary calculations, wearrive at

sin (X-1jJ) = ex :~ cos (fl-1J'). (113)

If we exclude variable 11' from this relationship for a fixed value of a by employing (112)and the fact that X = u + a, we arrive(since e = 0 at 'X = 0) at the differentialequation

~-1- sin u (114)dx - a (1-a cos u)

with the initial condition u (0) = 0, wherethe angle of lag plays the role of the soughtfunction. *

By substituting v for tan (u/2) in thedifferential equation (114), we can integratethe equation in closed form. However, theresulting u vs, 'X dependence proves to beextremely complicated, which, of course,hinders an effective study. Nevertheless,Eq. (114) can easily be integrated andstudied numerically. To this end we can

• See A.B. Tayler, "The sweep of a logging truck",Math. Spectrum 7, No.1: 19-26 (1974/75).

·126 Differential Equations in Applications

employ, for example, the Runge-Kutta second-order numerical method the essence ofwhich is the following.

Suppose that an integral curve u = 0) we select values U o, Ul' U 2 , •

such that u , ~ <Pi (X), where the successivevalues Ul' U 2 , are specified by theformulas

Ui+l == u., + (k1 + k2)/2,

with

kl = f (Xl' Ui) 8.X,

k 2 == t (Xi + fix, u, + k 1) /1X·

To solve Eq. (114) numerically with theinitial condition u (0) == 0, we compile thefollowing program using BASIC:

10 REM Runge-Kutta second-order method20 DEF FNF (X, U) =

1 - SIN(U)/(AL*(1 - A.COS(U»)30 CLS: PRINT "Solution of differential equa-

tion"40 PRINT "Runge-Kutta second-order method"100 PRINT "Parameters:"110 INPUT "Alpha=", AL120 INPUT "a=", A130 INPUT "Step of independent variable=", DX140 PRI NT "Initial values:"150 INPUT "of independent variable=", X


160 INPUT "of function==", U200 REM Next 20 values of function210 CLS: PRINT "X", "U"220 FOR I = 1 TO 20230 PRINT X, U240 K1 = DX*FNF (X,D)250 X = X + DX260 K2 = DX*FNF(X, U + K1)270 U == U + (K1 + K2)/2280 NEXT I290 INPUT "Continue (Yes = 1, No = 0)"; I300 IF I ( ) 0 GOTO 210310 END

Note that to compile the table of valuesof the solution for concrete values of a anda, we must know the limiting value C ofthis solution. The number C can be foundfrom the condition that

ex, (1 - a cos C) == sin C,

which leads to the formula

C - · -1 ( a, [1-a Yi.-a 2 + a2a2] )

- SIn 1+a2a2 •

Here, if a, == 1, we have C = n/2 2 tan-1 a, but if a, ~ 1, then C ~ a, (1 - a).The limiting value C can be approximatedby an exponential, namely

where 'V = a, (cos C - a)/sin2 C. For smalla's the value of 'V is fairly great and is


approximately given by the relationship

1y~ a(1-a)

But if a = 1, we have

a (1+a2 )

"(= 1-a2 •

For the step 6.X in the variation of theindependent variable X we must take anumber that does not exceed C. This requirement is especially important for small a'sand in the neighborhood of X = O. Belowwe give the protocol for calculating thevalues of the function u = u (X).

Solution of differential equation byRunge-Kutta second-order method

Parameters:alpha = 1.0a = 0.3Step of independent variable = 0.5Initial values:of independent variable = 0of Iunc tion = 0

xo.2.4.6.81

u

o.1437181.2299778.2824244.3146894.3347111


1.2 .34720881.4 .35504031.6 .359961.8 .36305562 .36500542.2 .36623432.4 .36700912.600001 .36749782.800001 .3678063.000001 .36800053.200001 .36812323.400001 .36820063.600001 .36824943.800001 .3682802Continue (Yes = 1, No = O)? 1

x u

4.000001 .36829974.200001 .3683124.4 .36831974.6 .36832464.8 .36832765 .36832965.2 .36833085.399999 .36833165.599999 .36833215.799999 .36833245.999999 .36833266.199999 .36833276.399998 .36833286.599998 .36833286.799998 .36833296.999998 .36833299-0770


7.199998 .36833297.399997 .36833297.599997 .36833297.799997 .3683329Continue (Yes =::::I: 1, No = O)? 1

X u

7.999997 .36833298.199997 .36833298.399997 .36833298.599997 .3683329B.799996 .36833298.999996 .36833299.199996 .36833299.399996 .36833299.599996 .36833299.799996 .36833299.999995 .368332910.2 .368332910.4 .368332910.6 .368332910.8 .368332910.99999 .368332911.t9999 .368332911.39999 .368332911.59999 .368332911.79999 .3683329Continue (Yes = 1, No = O)? 0OI(

The results of a numerical calculation ata = 0.3 are presented graphically in Figure 42. They show how an increase in X


influences the angle of lag of the loggingtruck. For clarity of exposition the scaleson the u and X axes are chosen to be different.

Now let us determine the sweep of thelogging truck by using the concept of halfwidth of the road at the outer curb of a turn(this quantity was defined earlier as the algebraic sum OX - OA + W (Figure 41)).First we note that

OX2 = ( ~ sin X-')..hCOSO)2

+( ; cos X+ ')..nsin 0)2

= h2 ( ~2 +')..2_ 2 ; sin u ) .

This shows that the sweep decreases as 'Xgrows since the angle of lag, u, increases.Thus, the maximal halfwidth ~h of theroad can be determined from the followingformula for ~:

V 1 1 WA= 'A2+---+-P a,1 a n :

In Figure 43 the solid curves reflect therelationship that exists between ~ - W/hand a for different values of A, while thedashed curves show what must be thevalue of ~ - L/h to guarantee the necessary"margin" at the inner curb of the turn. For


u

1

o

Fig. 42

C=D.9878826

c=a3683J29

C:eQ070fJ12-,-.._ ......_---~-~----

50 X

o

Fig. 43

1 ' "

Lfi-hf

o

(115)


every position of the logging truck on theroad we must have

p= ~ (i-cosu) + ~

< ~ (i-cos C) + ~

Next, since the value of C decreases with a,the angle of lag u proves to be the greatestwhen a = 0, which corresponds to the simplest case of logging. Then (115) yields

L 1 I f- Yi-a.2

p- - < - (1 - cos C) = .h a. a=O a

Now let us dwell on the results that followfrom the above line of reasoning. Here is atypical example that illustrates these results.

If a logging truck in which the distancebetween the front and back platforms is12 m follows a turn along an arc of a circleof radius 60 m, then ex = 0.2 and, in accordance with (109) N° is approximately 28°.If the width of the logging truck is 2.4 mand width of the load in the rear is 1.2 ill,

then for logs of approximately 24 m inlength (reckoning from the axis of thefront platform) we have A = 2, W/h =0.05, and L/h = 0.1. Thus, from Figure 43it follows that for all values of a we have~ = 0.45 for the outer curb of the turn and~ = 0.2 for the inner curb. Theoreticallythe necessary halfwidth of the road at the


outer curb of the turn is equal to 5.4 mand that at the inner curb to 2.4 m. If thelogging truck transports logs whose lengthis 14.4 m (reckoned from the axis of thefront platform) and whose bunch width inthe rear is 1.8 m, then in the case at handA == 1.2. The value of ~ then proves to bethe same for the inner and outer curbs ofthe turn and equal to 0.22. Hence, as caneasily be seen, the necessary halfwidth ofthe road at the outer curb of the turn is2.64 ill, while the same quantity at theinner curb, as in the previous case, is equalto 2.4 m. This reasoning shows that thelonger the logs transported the wider theroad at a turn must be. For one, if we compare the two cases considered here, we seethat an increase in the length of the logs by9.6 m requires widening the road at a turnby 2.76 m so that the driver can drive thetruck along a curve whose length at the turnis approximately equal to the length of theroad's center line. Practice has shown thatan inexperienced driver is not able todrive his truck along such a curve and needsa road whose total width at a turn is atleast 10.8 m (if the load transported is 24 mlong) for a truck width of 2.4 m.

The theory developed above shows thatthe sweep of the logging truck is the greatest when the truck enters the turn, sincein this case the angle of lag grows. Thisconclusion also holds true in the situation


when the truck enters one section of a zigzagturn after completing the previous one atthe point of inflection. The results represented graphically in Figure 43 correspond tothe case where prior to entering a turn thetractor unit and the trailer are positionedon a single straight line. But if there existsa nonzero initial angle of lag Cocaused bythe zigzag nature of the turn, we must selectthe initial condition in the differentialequation (114) in the form u (0) = -Co.Then the required width of the road isdetermined from the following formula for ~:

~= .. ;,.A2+ _1_+ 2 ~SillCo __1 +~V a2 ex. a 11,

"Ve note that in the case of simple logtransportation, that is, a=O, it is impossibleto pass a turn with ex greater than unity.However, for fairly large values of a thevalues ex > 1 become possible, they mustobey the following relationship: a (1 a cos C) == sin C. Thus, the maximal valueof ex is (1 - a2 ) -1/2, with the practicalextremal value of ex being 1.25 at a = 0.5.

We note in conclusion that for ex > 0.5considerable economy in the width of a roadis achieved by increasing a (see Figure 43).If the load is such that Ais not much greaterthan unity, the value of ex is chosen suchthat the required halfwidth of the road atthe inner curb of any turn is always smallerthan at the outer curb.

Chapter 2

Qualitative Methodsof Studying DifferentialModels

In solving the problems discussed in Chapter 1 we constructed differential modelsand then sought answers by integrating theresulting differential equations. However,as noted in the Preface, the overwhelmingmajority of differential equations are notintegrable in closed (analytical) form.Hence, to study differential models of realphenomena and processes we need methodsthat will enable us to extract the necessaryinformation from the properties of thedifferential equation proper. Below withconcrete examples we show how in solvingpractical problems one can use the simplestapproaches and methods of the qualitativetheory of ordinary differential equations.

2.1 Curves of Constant Directionof Magnetic Needle

Let us see how in qualitative integration,that is, the process of establishing the general nature of solutions to ordinary differential equations, one can use a general property of such equations whose analogue, for

Ch. 2. Qualitative Methods 137

(116)

example, is the property of the magneticfield existing at Earth's surface. The readerwill recall that curves at Earth's surfacecan be specified along which the directionof a magnetic needle is constant.

Thus, let us consider the first-order ordinary differential equation

dy (di =/ x, y),

where the function f (x, y) is assumed singlevalued and continuous over the set ofvariables x and y within a certain domain Dof the (x, y)-plane. To each point M (x, y)belonging to the domain D of functiont (x, y) the differential equation assignsa value of dy/dx, the slope K of the tangentto the integral curve at point M (x, y).Bearing this in mind, we say that at eachpoint M (z, y) of D the differential equation(116) defines a direction or a line element.The collection of all line elements in D issaid to be the field of directions or theline element field. Graphically a linear element is depicted by a segment for whichpoint M (z, y) is an interior point and whichmakes an angle ewith the positive directionof the x axis such that K = tan 8 =f (z, y). From this it follows that geometrically the differential equation (116) expressesthe fact that the direction of a tangent atevery point of an integral curve coincides withthe direction of the field (It this point.


To construct the field of directions it hasproved expedient to use the concept of isoclinals (derived from the Greek words for"equal" and "sloping"), that is, the set ofpoints in the (z, y)-plane at which thedirection of the field specified by the differential equation (116) is the same.

The isoclinals of the magnetic field atEarth's surface are curves at each point ofwhich a magnetic needle points in the samedirection. As for the differential equation(116), its isoclinals are given by the equationt (z, y) === v,

where 'V is a varying real parameter.Knowing the isoclinals, we can approx

imately establish the behavior of theintegral curves of a given differential equation. Let us consider, for example, thedifferential equation

:~ =x2+v,which cannot be integrated in closed (analytical) form. The form of this equationsuggests that the family of isoclinals is givenby the equati on

x2 + y2 = 'V, V > 0,

that is, the isoclinals are concentric circlesof radius VV- centered at the origin and lying in the (x, y)-plane, At each point of

Ch. 2. Qualitative Methods

a:

139

Fig. 44

such an isoclinal the slope of the tangentto the integral curve that passes throughthis point is equal to the squared radius ofthe corresponding circle. This informationalone is sufficient to convey an idea of thebehavior of the integral curves of thegiven differential equation (Figure 44).

We arrived at the final result quicklybecause the example was fairly simple.However, even with more complicated equations knowing the isoclinals may prove tobe expedient in solving a specific problem.

Let us consider a geometric method ofintegration of differential equations of the


type (116). The method is based on usingthe geometric properties of the curvesgiven by the equations

t (x, y) =0, (10)

of (x, y) + of (%, II) f ( ) - 0 (L)0% ay· x, y - •

Equation (10) , that is, the "zero" isoclinalequation, specifies curves at whose pointsdy/dx = O. This means that the points ofthese curves may prove to be points of maxima or minima for the integral curves ofthe initial differential equation. This explains why out of the entire set of isoclinalswe isolate the "zero" isoclinal.

For greater precision in constructingintegral curves it is common to find theset of inflection points of these curves (provided that SUCll points exist). As is known,points of inflection should be sought amongthe points at which y" vanishes. EmployingEq. (116), we find that

,,_ af(x, y) + 8f(x, y) ,Y - ax By Y

_ of (x, y) + of (x, y) f ( )ax By x, y ·

The curves specified by Eq. (L) are the possible point-of-inflection curves. * Note, for

• It is assumed here that integral curves thatfill a certain domain possess the property that onlyone integral curve passes through each point of thedomain.

Ch, 2. Qualitative Methods f41

one, that a point of inflection of an integralcurve is a point at which the integral curvetouches an isoclinal.

The:curves consisting of extremum points(maxima and minima points) and points ofinflection of integral curves break down thodomain of f into such subdomains 8 1 ,

8 2 , ., 8 m in which the first and secondderivatives of the solution to the differential equation have definite signs. Ineach specific case these subdomains shouldbe found. This enables giving a roughpicture of the behavior of integral curves.

As an example let us consider the differential equation y' = x + y. The equationof curve (I 0) in this case has the formx + y = 0, or y = -x. A direct checkverifies that curve (I 0) is not an integralcurve. As for curve (L), whose equation inthis case is y + x + 1 = 0, we find thatit is an integral curve and, hence, is nota point-of-inflection curve.

The straight lines (10) and (L) breakdown the (x, y)-plane into three sub domains(Figure 45): 8 1 (y' > 0, y" > 0), to theright of the straight line (10)' S2 (Y' < 0,y" > 0), between the straight lines (10) and(L), and 8 3 (y' < 0, y" < 0), to the left of thestraight line (L). The points of minima of theintegral curves lie on the straight line (10) .

To the right of (I 0) the integral curves pointupward, while to the left they point downward (left to right in Figure 45). There are

f42 Differential Equations in Applications

Fig. 45

no points of inflection. To the right of thestraight line (L) the curves are convex downward and, to the left convex upward. Thebehavior of the integral curves on the wholeis shown in Figure 45.

Note that in the given case the integralcurve (L) is a kind of "dividing" line, sinceit separates one family of integral curvesfrom another. Such a curve is commonlyknown as a separatrix.

Chi 21 Qualitative Methods

2.2 Why Must an Engineer KnowExistence and UniquenessTheorems?

143

When speaking of isoclinals and point-ofinflection curves in Section 2.1, we tacitlyassumed that the differential equation inquestion had a solution. The problem ofwhen a solution exists and of when it isunique is solved by the so-called existenceand uniqueness theorems. These theoremsare important for both theory and practice.

Existence and uniqueness theorems arehighly important because they guaranteethe legitimacy of using the qualitative methods of the theory of differential equationsto solve problems that emerge in scienceand engineering. They serve as a basis forcreating new methods and theories. Oftentheir proof is constructive, that is, the methods by which the theorems are proved suggest methods of finding approximate solutions with any degree of accuracy. Thus,existence and uniqueness theorems lie atthe base of not only the above-noted qualitative theory of differential equations butalso the methods of numerical integration.

Many methods of numerical solution ofdifferential equations have been developed,and although they have the common drawback that each provides only a concretesolution, which narrows their practicalpotential, they are widely used. It must be


noted, however, that before numericallyintegrating a differential equation one mustalways turn to existence and uniqueness theorems. This is essential to avoidmisunderstandings and incorrect conclusions.

To illustrate what has been said, let ustake two simple examples,* but first let usformulate one variant of existence anduniqueness theorems.

Existence theorem If in Eq. (116) function f is defined and continuous on a boundeddomain D in the (x, y)-plane, then for everypoint (xo, Yo) ED there exists a solution y (x)to the initial-value problem **

dy (dz = f x, y), (117)

that is defined on a certain interval containing point X o-

Existence and uniqueness theorem Ifin Eq. (116) junction f is defined and continuous on a bounded domain D in the (z, y)-

• See C.E. Roberts, Jr., "Why teach existence anduniqueness theorems in the first course of ordinarydifferential equations?", Int. J. Math. Educ. Sci.Technol. 7, No.1: 41-44 (1976).** If we wish to find a solution of a differentialequation satisfying a certain initial condition (inour case the initial condition is y (xo) = Yo), sucha problem is said to be an initial-value problem.

plane and satisfies in D the Lipschitz condition in variable y, that is,

I f (x, Y2) - f (x, Yl) I < L I Y2 - v, I,

uiith. L a positive constant, then for everypoint (xo, Yo) ED there exists a unique solution y (x) to the initial-value problem (117)defined on a certai-n interval containingpoint Xo.

Extension theorem If the hypotheses ofthe existence theorem or the .extstence anduniqueness theorem are satisfied, then everysolution to Eq. (116) with the initial data(xo, Yo) ED can be extended to a point thatlies as close to the boundary of D as desired.In the first case the extension is not necessarilyunique uihi le in the second it is.

Let us consider the following problem.Using the numerical Euler integrationmethod with the iteration scheme Yi+l =Yi+hf(xf, Yt) and step h=O.1, solvethe initial-value problem

y' = -x/y, y (- 1) = 0.21 (118)

on the interval [- 1, 3).Note that the problem involving the

equation y' £:: -x/y emerges, for example,from the problem considered on p. 162 concerned with a conservative system consisting of an object oscillating horizontally in10-0770

a vacuum under forces exerted by linearsprings.

To numerically integrate the initial-valueproblem (117) on the [-1, 2.8] intervalwe compile a program for building thegraph of the solution:

10 REM Numerical integration of differentialequation

20 DEF FNF(X, Y) =30 GOSUB 1110: REM Coordinate axes40 REM Next value of function1000 LINE -(FNX(X), FNY(Y», 21010 IF X < 2.9 GOTO 1001020 LOCATE 23, 11030' END1100 REM Construction of coordinate axes1110 SCREEN 1, 1, 0: KEY OFF: CLS1120 DEF FNX(X) = 88 + 80·X1130 DEF FNY(Y) = 96 - 80*Y1140 REM Legends on coordinate axes1150 LOCATE 1, 11, 0 PRINT "Y"1160 LOCATE 3, 11: PRINT "1"1170 LOCATE 13, 39: PRINT "X"1180 LOCATE 23, 10: PRINT u_1"1190 FOR 1 = -1 TO 2: LOCATE 13, 10.1 +

11: PRINT USING" # #"; I;: NEXT I1200 REM Oy1210 DRAW "BM88, ONM - 2, +8NM + 2,

+8D16"1220 FOR I = 1 TO 11: DRAW "NR2D16": NEXT1230 REM Ox


1240 DRAW "BM319, 96NM - 7, -2NM - 7+2L23"

1250 FOR I = 1 TO 19: DRAW "NU2L16": NEXT1260 REM Input of initial data1270 LOCATE 25, 1: PRINT STRING$(40, "),1280 LOCATE 25, t: INPUT, "xO =", X: INPUT;

"dx =", DX: INPUT; "yO =", y1290 PSET (FNX(X), FNY(Y»), 21300 RETURN

Here in line 20 the right-hand side of thedifferential equation is specified, and lines50 to 990 must hold the program of thenumerical integration of the differentialequation.

In our case (the initial-value problem(118») the beginning of the program hasthe following form:

10 REM Euler's method~O DEF FNF(X, Y) = -X/Y30 GOSUB 1100: REM Coordinate axes40 REM Next value of function100 y = Y + DX*FNF(X, Y)110 X = X + DX

The results are presented graphically inFigure 46.

Let us now turn to the existence theorem.For the initial-value problem (118), thefunction f (x, y) = -x/y is defined andcontinuous in the entire (x, y)-plane exclud-10*

<.0~.

~

C::/

~ co C'.J ~ co ~

c::::J r::::J t:::J" c::::; I~I Ic:::1

I

coc::)I

co~

...... tlOI rz

ing the x axis. Thus, in accordance withthe existence theorem, the initial-valueproblem (118) has a solution y (x) defined ona certain interval containing point X o =-1. This solution, according to the extension theorem, can be extended to a valueof y (x) close to the value y (x) = O. Asa result of numerical integration we havearrived at a solution of (118) defined on aninterval (a, b), with a < -1 and 1.3 <b< 1.4. However, allowing for the concreteform of the differential equation, we canspecify the true interval in which thesolution to the initial-value problem (118)exists. Indeed, since in the initial equationthe variables can be separated, we have

y x

~ 11 dll = - ~ ~ d~.0.21 -1

Integrating, we get y = V 1.0441 - x2:

Hence, a solution to the initial-value prob-lem (118) exists only for I x I < V1.0441~ 1.0218.

Thus, by resorting to the existence theorem (and to the extension theorem) we wereable to "cut off" the segment on which thereis certain to be no solution of the initialvalue problem. If we employ only numericalintegration, we arrive at an erroneousresult. TIle fact is that as the solutionY = Y (x) approaches the x axis, the angle


of slope of the curve tends to 90°. Therefore,in the time that the independent variable x changes by 0.1, the value of y is able to"jump over" the x axis, and we find ourselveson an integral curve that differs from theoriginal. This happens because the Eulermethod allows for the angle of slope onlyat the running point.

The following example is even moreinstructive. We wish to solve the initialvalue problem

»r:y' = 3xy y, Y (-1) = -1 (119)

on the segment [- 1, 1]. The approach hereconsists in first employing the Euler methodand then an improved Euler method" witha step h == 0.1 and an iteration schemeYi+l == Yi + hj (Xi+1/2' Yi+1/2)' withYi+1/2 == Yi + hj (Xb Yi)/2.

We solve the initial-value problem (119)using the above program, whose beginningin the case at hand has the following form:

10 REM Euler's method20 DEF FNF(X, Y) == 3.X.SGN(Y)*ABS(Y) ---.

(1/3)30 GOSUB 1100: REM Coordinate axes40 REM Next value of function100 y == Y + DX*FNF(X, Y)110 X == X + DX

The results are presented graphically illFigure 47.


0.6

0.2

-0.2 0 QZ-Q2

-an

- 1

as

Fig. 47

As for the improved Euler method, thebeginning of the program has the followingform:

10 REM Improved Euler's method20 DEF FNF(X, Y) = 3.X*SGN(Y)*ABS(Y) "

(1/3)30 GOSUB 1100: REM Coordinate axes40 REM Next value of function50 D2 = DX/2


y1

Q6

-1 -0.0

Fig. 48

100 Y1 = Y + D2*FNF(X, Y)110 Y = Y + DX.FNF(X + D2, Y1)120 X = X + DX

a:

The results are presented graphically inFigure 48, and the diagram differs fromthat depicted in Figure 47.

To look into the reason for SUCll strikingdiscrepancy in the results, we integrate the


initial-value problem. Separating the variables, we get

y x

~ 11-1/

3 d1') = 3 ~ ~ d~,-1 -1

Of, finally, Y = ±x3• This result alreadysuggests that the solution via the Eulermethod gives the function Yt (x) = x3 whilethe solution via the improved Euler methodgives

x3 if x~O,

-x3 if x> O.

Both Yl and Y2 are solutions to the initialvalue problem (119), which means that thesolution of the initial-value problem considered on the segment [-1, 1] is notunique.

Let us now turn to the existence anduniqueness theorem in connection with thisproblem. First, we note that since the func-tion f (x, y) = 3xVy is continuous in theentire (x, y)-plane, the existence theoremimplies that the initial-value problem (119)has a solution defined on a segment containing point X o = -1, and, according tothe extension theorem, this solution canbe extended to any segment. Further, sinceof (z, y)/8y = xy-2/3, the function f (x, y)=


3xV'Y satisfies the Lipschitz condition invariable y in any domain not containingthe x axis. If, however, a domain doescontain points belonging to the x axis, itis easy to show that the function doesnot satisfy the Lipschitz condition. Hence,from the existence and uniqueness theorem(and the extension theorem) i.t follows thatin this case the solution to the initial-valueproblem can be extended in a unique manner at least to the x axis. But since thestraight line y = 0 constitutes a singularintegral curve of the differential equationy' = 3xVY:- we already know that as soonas y vanishes, there is no way in which wecan extend the solution to the initial-valueproblem (119) beyond point 0 (0, 0) ina unique manner.

Thus, by resorting to the existence anduniqueness theorem (and the extensiontheorem) we were able to understand theresults of numerical integration, that is, ifwe are speaking of the uniqueness of thesolution of the initial-value problem (119)on the [-1, 1] segment, the solution existsand is defined only on the segment [-1, 0].Generally, however, there can be severalsuch solutions.


2.3 A Dynamical Interpretationof Second-Order DifferentialEquations

Let us consider the nonlinear differentialequation

d2x ( dX)

dt 2 =1 x,dt (120)

whose particular case is the second-order differential equation obtained on p. 72 whenwe considered pendulum clocks. We takea simple dynamical system consisting ofa particle of unit mass that moves along thex axis (Figure 49) and on which a forcef (x, dx/dt) acts. Then the differentialequation (120) is the equation of motionof the particle. The values of x and dx/dtat each moment in time characterize thestate of the system and correspond to apoint in the (x, dx/dt)-plane (Figure 50),which is known as the plane of states orthe phase (x, dx/dt)-plane. The phase planedepicts the set of all possible states of thedynamical system considered. Each new stateof the system corresponds to a new point inthe phase plane. Thus, the changes in thestate of the system can be represented bythe motion of a certain point in the phaseplane. This point is called a representativepoint, the trajectory of the representativepoint is known as the phase trajectory,


Fig. 49

•o

( ax)f,x'{l[~

•cZ'

dxdt

o

Fig. 50

a:

and the rate of motion of this point as thephase velocity.

If we introduce the variable y = dxldt,Eq. (120) can be reduced to a system oftwo differential equations:

dx d!ldt = Y, dt = / (x, y). (121)

If we take t as a parameter, then the solution to system (121) consists of two functions, x (t) and y (t), that in the phase(z, y)-plane define a curve (a phase trajectory).


It can be shown that system (121) andeven a more general system

~~ = X (x, y), ~; = y (x, y), (122)

where the functions X and Y and theirpartial derivatives are continuous in a domain D, posseses the property that if x (t)and y (t) constitute a'solution to the differential system, we can write

x = x (t + C), y = Y (t + C), (123)

where C is an arbitrary real constant, and(123) also constitute a solution to the samedifferential system. All solutions (123)with different values of C correspond to asingle phase trajectory in the phase (z, y)plane. Further, if two phase trajectorieshave at least one common point, theycoincide. Here the increase or decrease inparameter t corresponds to a certain direction of motion of the representative pointalong the trajectory. In other words, aphase trajectory is a directed, or oriented,curve. When we are interested in thedirection of the curve, we depict the direction of the representative point along thetrajectory by placing a small arrow on thecurve.

Systems of the (122) type belong to theclass of autonomous systems of differentialequations, that is, systems of ordinarydifferential equations whose right-hand


sides do not explicitly depend on time t.But if at least in one of the equations ofthe system the right-hand side dependsexplicitly on time t, then such a system issaid to be nonautonomous.

In connection with this classification ofdifferential equations the following remarkis in order. If a solution x (t) to Eq. (120)is a nonconstant periodic solution, thenthe phase trajectory of the representativepoint in the phase (x, y)-plane is a simpleclosed curve, that is, a closed curve withoutself-intersections. The converse is also true.

If differential systems of the (122) typeare specified in the entire (z, y)-plane, then,generally speaking, phase trajectories willcompletely cover the phase plane withoutintersections. And if it so happens that

X (xo, Yo) == y (xo, Yo) == 0

at a point M o(xo, Yo), the trajectory degenerates into a point. Such points are calledsingular. In what follows we consider primarily only isolated singular points. A singular point M 0 (xo, Yo) is said to be isolatedif there exists a neighborhood of this pointwhich contains no other singular pointsexcept M 0 (xo, Yo).

From the viewpoint of a physical interpretation of Eq. (120), the point M 0 (xo, 0)is a singular point. At this point y = 0 andt (xo, 0) = O. Thus, in this case the isolatedsingular point corresponds to the state of

Chi 2. Qualitative Methods 159

a particle of unit mass in which both thespeed dx/dt and the acceleration dy/dt ==d2x/dt2 of the particle are simultaneously zero, which simply means that the particleis in the state of rest or in equilibrium.In view of this, singular points are alsocalled points of rest or points of equilibrium.

The equilibrium states of a physical system constitute very special states of thesystem. Hence, a study of the types ofsingular points occupies an important placein the theory of differential equations.

The first to consider in detail the classification of singular points of differentialsystems of the (122) type was the distinguished Russian scientist Nikolai E. Zhukovsky (1847-1921). In his master's thesis "The kinematics of a liquid body",presented in 1876, this problem emerged inconnection with the theory of velocitiesand accelerations of fluids. The modernnames of various types of singular points weresuggested by the great French mathematician Jules H. Poincare (1854-1912).

Now let us try to answer the question ofthe physical meaning that can be attachedto phase trajectories and singular points ofdifferential systems of the (122) type. Forthe sake of clarity we introduce a twodimensional vector field (Figure 51) definedby the function

v (x, y) = X (z, y)i + Y (x, y)j,


Fig. 51

where i and j are the unit vectors directedalong the x and y axes, respectively, ina Cartesian system of coordinates. At everypoint P (x, y) the field has two components,the horizontal X (x, y) and the verticaly (x, y). Since dx/dt == X (x, y) and dy/dt=y (x, y),. the vector associated with eachnonsingular point P (x, y) is tangent atthis point to a phase trajectory.

If variable t is interpreted as time, vector V can be thought of as the vector ofvelocity of a representative point movingalong a trajectory. Thus, we can assumethat the entire phase plane is filled withrepresentative points and that each phasetrajectory constitutes the trace of a movingrepresentative point. As a result we arriveat an analogy with the two-dimensionalmotion of an incompressible fluid. Here,since system (122) is autonomous, vector V

Ch.. 2. Qualitative..Methods t6t

at each fixed point P (z, y) is time.-independent and, therefore, the motion of thefluid is steady-state. The phase trajectoriesin this :case are simply the trajectories ofthe moving particles of fluid and the singular points 0,0', and 0" (see Figure 51) arethose where the fluid is at rest.

The most characteristic features of thefluid motion shown in Figure 51 are (1)the presence of singular points, (2) thedifferent patterns of phase trajectories nearsingular points, (3) the stability or instability at singular points (i.e, two possibilities may realize themselves: the particlesthat are in the vicinity of singular pointsremain there with the passage of time orthey leave the vicinity for other parts ofthe plane), and (4) the presence of closedtrajectories, which in the given case correspond ~to periodic motion.

These features constitute the main partof the phase portrait, or the completequalitative behavior pattern offhe phasetrajectories of a general-type system (122).Since, as noted earlier, differential equations cannot generally be solved analytically,the aim of the qualitative theory of ordinarydifferential equations of the (122) type is tobuild a phase portrait as complete as possible directly from the functions X (x, y)and Y (z, y).

11-0770

1'62. Differential Equations in Applications

2.4 Conservative Systemsin Mechanics

Practice gives us ample examples of thefact that. any real dynamical systemdissipates energy. The dissipation usuallyoccurs as a result of some form of friction.But in some cases it is so slow that it can beneglected if the system is studied over a fairly small time interval. The law of energyconservation, namely, that the sum of kinetic and potential energies remains constant,can .be assumed to hold true for such systems. Systems of this kind are called conservative. For example, rotating Earth maybe seen as a conservative system if we takea time interval of several centuries. But ifwe study Earth's motion over several million years, we must allow for energy dissipation related to tidal flows of water in seasand oceans.

A simple example of a conservative system is one consisting of an object movinghorizontally in a vacuum under forces exerted by two springs (Figure 52). If x is thedisplacement of the object (mass m) fromthe state of equilibrium and the force withwhich the two springs act on the object (therestoring force) is proportional to x, theequation of motion has the form

d2xm (fii"" +kx = 0, k ;» O.


t OOOOOO~OOOOO ±uoooo~oooooo jFig. 52

Springs of this type are known as linear,since the restoring force exerted by them isa linear function of x.

If an object of mass m moves in a medium that exerts a drag on it (the dampingforce) proportional to the object's velocity,the equation of motion for such a nonconservative system is

d2x dzm""""(j}2+cdt+kx=O, c>o. (124)

Here we are dealing with linear damping,since the damping force is a linear functionof velocity dx/dt.

If f and g are such arbitrary functions thatf (0) = 0 and g (0) = 0, the more generalequation,

m ::~ +g (~~ )+f(x)=O, (125)

can be interpreted as the equation ofmotion of an object of mass m undera restoring force -f (x) and a dampingforce -g (dx/dt). Generally, these forces

11*


are nonlinear; hence Eq. (125) can be con...sidered the basic equation of nonlinear mechanics.

Let us hriefly examine the special case ofa nonlinear conservative system describedby the equation

d2xm (lt2 + f (x) = 0, (126)

where the damping force is zero and, hence,energy is not dissipated. From Eq. (126) wecan pass to the autonomous system

dx ~__ f(x)dt=Y, dt - m (127)

If we now exclude time t from Eqs. (127),we arrive at a differential equation for thetrajectory of the system in the phase plane:

~__ f(x)dx - my

This equation can be written as

mydy= -f(x) dx.

(128)

(129)

Then, assuming that x = Xo at t = to andy = Yo, we can integrate Eq. (129) from toto t. The resul t is

x

~ my2_+ my~= - ~ f (~) d~,~o


which may be rewritten as

165

(131)

x X o

+myLI- ~ f(s)ds=+my:+ ~ f(x)dx.o 0

(130)

Note that my2/2 == In (dxldt)2/2 is the kinetic energy of a dynamical system and

x

V (x) = ~ f(~) dso

is the system's potential energy. Thus,Eq. (130) expresses the law of energy censervation:

12" my2+V (x) ==E, (132)

where E == lny~!2 + V (xo) is the total energy of the system. Clearly, Eq. (132) is theequation of the phase trajectories of system (127), since it is obtained by integratingEq. (128). Thus, different values of E correspond to different curves of constant energyin the phase plane. The singular points ofsystem (127) are the points M v (xv, 0),where x; are the roots of the equation f (x) =O. As noted earlier, the singular points arepoints of equilibrium of the dynamical system described by Eq. (126). Equation (128)implies that the phase trajectories of thesystem intersect the x axis at right angles,while the straight lines x ::;:;: Xv are parallel


to the x axis. In addition, Eq. (132) showsthat the phase trajectories are symmetricwith respect to the x axis.

In this case, if we write Eq. (132) in theform

y = + 11 ~ [E - V (x)] , (133)

we can easily plot the phase trajectories.Indeed, let us introduce the (x, z)-plane,the plane of energy balance (Figure 53),with the z axis lying on the same verticalline as the y axis of the phase plane. We thenplot the graph of the function z = V (x) andseveral straight lines z == E in the (x, z)plane (one such straight line is depicted inFigure 53). We mark a value of E - V (x)on the graph. Then for a definite x we multiply E - V (x) by 21m and allow for formula (133). This enables us to mark the respective values of y in the phase plane. Notethat since dx/dt = y, the positive directionalong any trajectory is determined by themotion of the representative point from leftto right above the x axis and from right toleft below the x axis.

The above reasoning is fairly general andmakes it possible to investigate the equation of motion of a pendulum in a mediumwithout drag, which has the form (see p. 72)

d2x .dJ2 +k SIn x ~ 0, (134)


z

Ol-- -+--__t--_~

o:

!/

.Xo~---......::::::...~---+---r---~

Fig. 53

with k a positive constant.Since Eq. (134) constitutes a particular

case of Eq. (126), it can be interpreted asan equation describing the frictionless mo-.tion in a straight line of an object of uni tmass under a restoring force equal to


-k sin x. In this case the autonomoussystem corresponding to Eq. (134) is

dx dy .dt=Y, (It = -kSIDX. (135)

The singular points are (0, 0), (± n, 0),(± 2,,;, 0), .. , and the differential equation of the phase trajectories of system (135)assumes the form

dy _ ksinx(iX---y-.

Separating the variables and integrating,we arrive at an equation for the phase trajectories,

t2 y2:.+ k (1 - cos x) = E.

This equation is a particular case ofEq. (132)with m = 1, where the potential energydetermined by (131) is specified by therelationship

x

V(x) = ~ f(s)ds=k(1-cosx).o

In the (x, z)-plane we plot the functionz = V (x) as well as several straight linesz = E (in Figure 54 only one such line,z = E = 2k, is shown). After determininga value of E - V (x) we can draw a sketch

'Ch. 2. Qualitative Methods 169

Fig. 54

of the trajectories in the phase plane byemploying the relationship

Y= ± Y2 [E- V (x»).


The resulting phase portrait shows (seeFigure 54) that if the energy E varies from 0to 2k, the corresponding phase trajectoriesprove to be closed and Eq. (134) acquires.periodic solutions. On the other hand, ifE > 2k, the respective phase trajectoriesare not closed and Eq. (134) has no periodicsolutions. Finally, the value E = 2k corresponds to a phase trajectory in the phaseplane that separates two types of motion,that is, is a separatrix. The wavy lineslying outside the separatrices correspond torotations of a pendulum, while the closedtrajectories lying inside the regions bounded by separatrices correspond to oscillations of the pendulum.

Figure 54 shows that in the vicinity ofthe singular points (+2nm, 0), In = 0, 1,2 t •• , the behavior of the phase trajectories differs from that of the phase trajectories in the vicinity of the singular points(+nn, 0), where n = 1, 2, .

There are different types of singular points.With some we will get acquainted shortly.As for the above example, the singularpoints (± 2Jtm, 0), m = 0, 1, 2, ... , belong to the vortex-point type, while thesingular points (+nn, 0), n = 1, 2, .. ,belong to the saddle-point type. A singularpoint of an autonomous differential systemof the (122) type is said to be a vortexpoint jf there exists a neighborhood of thispoint completely filled with nonintersecting


phase trajectories surrounding the point.A saddle point is a singular point adjoinedby a finite number of phase trajectories("whiskers") separating a neighborhood ofthe singular point into regions where thetrajectories behave like a family of hyperbolas defined by the equation xy = const.

Now let us establish the effect of linearfriction on the behavior of the· phase trajectories of a conservative system, Theequation isd2x dx .(ft"2+ cdt+ k SID X =0, c:» O.

If friction is low, that is, the pendulum isable to oscillate about the position ofequilibrium, it can be shown that th-e" .pli-asetrajectories arc such as shown in Figure 55.But if friction is so high that oscillationsbecome impossible, the pattern of phasetrajectories resembles the one depicted inFigure 56.

If we now compare the phase portrait ofa conservative system with the last two portraits of nonconservative systems, we seethat saddle points have not changed (weconsider only small neighborhoods of singular points), while in the neighborhood ofthe singular points (±21tm, 0), m = 0, 1,2, .. , the closed phase trajectories havetransformed into spirals (for low friction)or into trajectories that "enter" the singularpoints in certain directions (for high fric-

174 Differential Equations in Applica tions

tion). In the first case (spirals) we havesingular points of the focal-point type andin the second, of the nodal-point type.

A singular point of a two-dimensionalautonomous differential system of the general type (122) (if such a point exists)is said to be a focal point if there existsa neighborhood of this point that is completely filled with nonintersecting phasetrajectories resembling spirals that "wind"onto the singular point either as t-e- +00or as t--+ -00. A nodal point is a singularpoint in whose neighborhood each phasetrajectory behaves like a branch of a parabola or a half-line adjoining the point alonga certain direction.

Note that if a conservative system hasa periodic solution, the solution cannot beisolated. More than that, if r is a closedphase trajectory corresponding to a periodicsolution of the conservative system, thereexists a certain neighborhood of r that iscompletely filled with closed phase trajectories.

Note, in addition, that the above definitions of types of singular points havea purely qualitative, descriptive nature.As for the analytical features of these types,there are no criteria, unfortunately, inthe general case of systems of the (122)types, but for some classes of differentialequations such criteria can be form ulated.


A simple example is the linear system

dx dy"dT = atx +bty, dt = a2x +b2y,

where aI' bl , a 2, and b2 are real constants.If the coefficient matrix of this sytem is

nonsingular, that is, the determinant

the origin 0 (0, 0) of the phase plane isthe only singular point of the differentialsystem.

Assuming the last inequality valid, we denote the eigenvalues of the coefficient m·atrix by Al and A2• It can then be demonstrated that

(1) if Al and A2 are real and of the samesign, the singular point is a nodal point,

(2) if Al and A2 are real and of oppositesign, the singular point is a saddle point,

(3) if A] and A2 are not real and are notpure imaginary, the singular point is a focal point, and

(4) if Al and A2 are pure imaginary, thesingular point is a vortex point.

Note that the first three types of singularpoints belong to the so-called coarse singular points, that is, singular points whosenature is not affected by small perturbations of the right-hand sides of the initialdifferential system.. On the other hand,


a vortex point is a fine singular point; itsnature changes even under small perturbations of the right-hand sides.

2.5 Stability of Equilibriwn Pointsand of Periodic Motion

As we already know, singular points ofdifferent types are characterized by differentpatterns of the phase trajectories in sufficiently small neighborhoods of these points. Thereis also another characteristic, the stabilityof an equilibrium point, which providesadditional information on the behavior ofphase trajectories in the neighborhood ofsingular points. Consider the pendulum depicted in Figure 57. Two states of equilibrium are shown: (a) an object of mass m isin a state of equilibrium at the uppermostpoint, and (b) the same object is in a stateof equilibrium at its lowest point. Thefirst state is unstable and the second, stable.And this is why. If the object is in itsuppermost state of equilibrium, a slightpush is enough to start it moving with anever increasing speed away from the equilibrium position and, hence, away from theinitial position. But if the object is in thelowest possible state, a push makes it moveaway from the position of equilibriumwith a decreasing speed, and the weaker thepush the smaller the distance by which the

Ch. 2. Qualitative Methods,IIIIIII

Fig. 57

177

object is displaced from the initial position.

The state of equilibrium of a physicalsystem corresponds to a singular point inthe phase plane. Small perturbations at anunstable point of equilibrium lead to largedisplacements from this point, while ata stable point of equilibrium small perturbations lead to small displacements. Starting from these pictorial ideas, let usconsider an isolated singular point of system (122), assuming for the sake of simplicity that the point is at the origin 0 (0, 0)of the phase plane. We will say that thissingular point is stable if for every posi tive Rthere exists a positive r~ R such that

12-0770

tSO Differential Equations in Applications

Nikolai G. Chetaev (1902-1959) wrote: *.v.If a passenger plane is being designed, acertain degree of stability must be providedfor in the future movements of the plane sothat it will be stable in flight and accidentfree during take-off and landing. The crankshaft must be so designed that it does notbreak from the vibrations that appear in realconditions of motor operation. To ensure thatan artillery gun has the highest possible accuracy of aim and the smallest possible spread,the gun and the projectiles must be constructed in such a manner that the trajectoriesof projectile flight are stable and the projectiles fly correctly.

Numerous examples can be added to thislist, and all will prove that real movementsrequire selecting out of the possible solutionsof the equations of motion only those thatcorrespond to stable states. Moreover, if wewish to avoid a certain solution, it is advisableto change the design of the correspondingdevice in such a way that the state of motioncorresponding to this solution becomes unstable.

Returning to the pendulum depicted inFigure 57, we note the following curiousand somewhat unexpected fact. Researchhas shown that the upper (unstable) position of equilibrium can be made stable by

• See N .G. Chetaev, Stability oj Motton (Moscow:Nauka, 1965: pp. 8-9 (in Russian).


introducing vertical oscillations of thepoint of suspension. More than that,not only the upper (vertical) position ofequilibrium can be made stable but also anyother of the pendulum's positions (for one,a horizontal position) by properly vibratingthe point of suspension. *

Now let us turn to a concept no less important than the stability of an equilibriumpoint, the concept of stability of periodicmovements (solutions). Let us assume thatwe are studying a conservative system thathas periodic solutions. In the phase planethese solutions are represented by closedtrajectories that completely fill a certainregion. Thus, to each periodic motion ofa conservative system there correspondsa motion of the representative point alonga fixed closed trajectory in the phase plane.

Generally, the period of traversal 0.£different trajectories by representativepoints is different. In other words, the periodof oscillations in a conservative systemdepends on the initial data. Geometricallythis means that two closely spaced representative points that begin moving ata certain moment t == to (say, at the x axis)will move ap art to a certain finite distance

* The reader call find many cxarr plr s cf stal.il ization of different t)PCS (If pr ndul ums in 11'r bookby T.G. Strizhak, Methods oj Investigating Tujnamical Systems of the "Pendulum" Type (Alrna-Ata:Nauka, 1981) (in Hussian).


Fig. 58

every phase trajectory originating at theinitial moment t == to at a point Plyinginside the circle x2 + y2 = r2 will lieinside the circle x2 + y2 = R2 for all t > to(Figure 58). Without adhering to rigorousreasoning we can say that a singular pointis stable if all phase trajectories that arenear the point initially remain there withthe passage of time. Also, a singular pointis said to be asymptotically stable if it isstable and if there exists a circle x2 +y2 = r: such that each trajectory that attime t = to lies inside the circle convergesto the origin as t~ +00. Finally, ifa singular point is not stable, it is said tobe unstable.

Ch. 2. Qualitative Methods f79

A singular point of the vortex type isalways stable (but not asymptotically).A saddle point is always unstable. InFigure 55, which illustrates the behaviorof the phase trajectories for pendulum oscillations in a medium with low drag, the singular points, focal points, are asymptotically stable; in Figure 56 the singular points,which are nodal points, are also asymptotically stable.

The introduced concept of stability of anequilibrium point is purely qualitative,since no mention of properties referring tothe behavior of phase trajectories has beenmade. As for the concept of asymptoticstability, if compared with the notion ofsimple stability, it is additionally necessary that every phase trajectory tend tothe origin with the passage of time. However, in this case, too, no conditions areimposed on how the phase trajectory mustapproach point 0 (0, 0).

The concepts of stability and asymptoticstahility play an important role in applications. The fact is that if a device isdesigned without due regard for stabilityconsiderations, when built it will be sensitive to the very smallest external perturbations, which in the final analysis may leadto extremely unpleasant consequences. Emphasizing the importance of the concept ofstability, the well-known Soviet specialistin the field of mathematics and mechanics

12·

{SO Differential Equations in Applications

Nikolai G. Chetaev (1902-1959) wrote: *... If a passenger plane is being designed, acertain degree of stability must be providedfor in the future movements of the plane sothat it will be stable in flight and accidentfree during take-off and landing. The crankshaft must be so designed that it does notbreak from the vibrations that appear in realconditions of motor operation. To ensure thatan artillery gun has the highest possible accuracy of aim and the smallest possible spread,the gun and the projectiles must he constructed in such a manner that the trajectoriesof projectile flight are stable and the projectiles fly correctly.

Numerous examples can be added to thislist, and all will prove that real movementsrequire selecting out of the possible solutionsof the equations of motion only those thatcorrespond to stable states. Moreover, if wewish to avoid a certain solution, it is advisableto change the design of the correspondingdevice in such a way that the state of motioncorresponding to this solution becomes unstable.

Returning to the pendulum depicted inFigure 57, we note the following curiousand somewhat unexpected fact. Researchhas shown that the upper (unstable) position of equilibrium can be made stable by

• See N.G. Chetaev, Stability of Matton (Moscow:Nauka, 1965: pp. 8-9 (in Russian).


in troducing vertical oscillations of thepoint of suspension. More than that,not only the upper (vertical) position ofequilibrium can be made stable but also anyother of the pendulum's positions (for one,a horizontal position) by properly vibratingthe point of suspension. *

Now let us turn to a concept no less important than the stability of an equilibriumpoint, the concept of stability of periodicmovements (solutions). Let us assume thatwe are studying a conservative system thathas periodic solutions. In the phase planethese solutions are represented by closedtrajectories that completely fill a certainregion. Thus, to each periodic motion ofa conservative system there correspondsa motion of the representative point alonga fixed closed trajectory in the phase plane.

Generally, the period of traversal ofdifferent trajectories by representativepoints is different. In other words, the periodof oscillations in a conservative systemdepends on the initial data. Geometricallythis means that two closely spaced representative points that begin moving ata certain moment t = to (say, at the x axis)will move apart to a certain finite distance

* The reader can find many cxarnplr s cf stal.il ization of differcnt t)"PCS (If pendulums in tho bookby T.G. Strizhak, Methods of Investigating Dynamical Systems of the "Pendulum" Type (Alma-Ata:Nauka, 1981) (in Russian).

182 Dlfferential Equations in Applications

Fig. 59

with the passage of time. But it also mayhappen that these points do not separate.To distinguish between these two possibilities, the concept of stability in the senseof Lyapunov is introduced for periodicsolutions. The essence of this concept liesin the following. If knowing an e-neighhorhood (with e as small as desired) of a pointM moving along a closed trajectory r(Figure 59) * ensures that we: know a moving B (e)-neighborhood of the same point Msuch that every representative point thatinitially lies in the B (ej-neighborhood willnever leave the s-neighborhood with thepassage of time, then the periodic solution

• An s-neighborhood of a point M is understoodto be a disk of radius e centered at point M,

Ch, 2. Qualitative Methods f83

corresponding to r is said to be stable inthe sense of Lyapunov. If a periodic solutionis not stable in the sense of Lyapunov, it issaid to be unstable in the sense of Lyapunov.

When it comes to periodic solutions thatare unstable in the sense of Lyapunov, wemust bear in mind that they still possesssome sort of stability, orbital stability;which means that under small variations ofinitial data the representative point transfers from one phase trajectory to anotherlying as close as desired to the initiallyconsidered trajectory.

Examples of periodic solutions that arestable in the sense of Lyapunov are thosethat emerge, for instance, when we considerthe differential equation that describes thehorizontal movements of an object of massm in a vacuum with two linear springs actingon the object (Figure 52). An example ofperiodic solutions that are unstable in thesense of Lyapunov but are orbitally stableis the solutions of the differential equation (134), which describes the motion ofa circular pendulum in a medium withoutdrag.

In the first case the oscillation perioddoes not depend on the initial data and isfound by using the formula T = 2n¥m/k.In the second the oscillation period dependson the initial data and is expressed, as weknow, in terms of an elliptic integral of thefirst kind taken from 0 to ~/2t


Finally, we note that the question ofwhether periodic movements are stable inthe sense of Lyapunov is directly linkedto the question of isochronous vibrations. *

2.6 Lyapunov Functions

Intuitively it is clear that if the totalenergy of a physical system is at its minimum at a point of equilibrium, the pointis one of stable equilibrium. This idea liesat the base of one of two methods used instudying stability problems, both suggestedby the famous Russian mathematician Aleksandr M. Lyapunov (1857-1918). This method is known as Lyapunov's direct, orsecond, method for stability investigations. **

We illustrate Lyapunov's direct methodusing the (122) type of system when theorigin is a singular point.

Suppose that r is a phase trajectory ofsystem (122). We consider a function V =V (z, y) that is continuous together with itsfirst partial derivatives BV/Bx and BV/By

• See, for example, the book by V. V. ArneI' kin,N. A. Lukashevich, and A.P. Sadovskii, N onlinear Vibrations in Second-Order Systems (Minsk:Belorussian Univ. Press, 1982) (in Russian).*. The reader will find many interesting examplesof stability investigations involving differentialmodels in the book by N. Rouche, P. Habets, andM. Laloy, Stability Theory by Lyapunov's DirectMethod (New York: Springer, 1977).


in a domain containing r in the phase plane.If the representative point (x (t), y (z)moves along curve r, then on this curvethe function V (z, y) may be considereda function of t, with the result that therate at which V (x, y) varies along r isgi ven by the formula

~-~ i=.+.~ ~-~X(x y)dt - ox dt oy dt - ax '

av+ay y (x, y), (136)

where X (z, y) and Y (z, y) are the righthand sides of system (122).

Formula (136) is essential in the realization of Lyapunov's direct method. Thefollowing concepts are important for thepractical application of the method.

Suppose that V = V (x, y) is continuous together with its first partial derivatives av/ax and av/ay in a domain G containing the origin in the phase plane, withV (0, 0) = O. This function is said to bepositive (negative) definite if at all pointsof G except the origin V (x, y) it is positive(negative). But if at points of G we haveV (x, y)~ 0 (~O), the function V = V (z,y) is said to be nonnegative (nonpositive).For example, the function V defined by theformula V (x, y) = x2 + y2 and consideredin the (x, y)-plane is positive definite, whilethe function V (x, y) = x2 is nonnegativesince it vanishes on the entire y axis.

(137)


If V (x, y) is positive definite, we canrequire that at all points of the domainG,O the following inequality hold true:

( ~~ t+( ~: t~O.This means that the equation z = V (x, y)can be interpreted as the equation of a surface resembling a paraboloid that touchesthe (x, y)-plane at point 0 (0, 0) (Figure60). Generally, the equation z == V (x, y)with a positive definite V may specifya surface of a more complex structure. Onesuch surface is shown in Figure 61, wherethe section of the surface with the planez == C results not in a curve but in a ring.

If a positive definite function V (x, y) issuch that

W (x, y) = DV ~~ y) X (x, y)

+ avo~' y) y (x, y)

is nonpositive, V is said to be the Lyapunovfunction of system (122). We note here thatin view of (136) the requirement that W benonpositive means that dVldt~ 0 and,hence, the function V == V (x, y) does notincrease along the trajectory r in theneighborhood of the origin.

Here is a result arrived at by Lyapunov:if for system (122) there exists a Lyapunov[unction V (x, 1I)~ then the origin; which is


z

x

Fig. 60

187

Fig. 61

a singular point, is stable. If the positivedefinite function V = V (x, y) is such thatfunction W defined via (137) is negative definite, then the origin is asymptotically stable.

We will show with an example. how tQ


apply the above result. Let us consider theequation of motion of an object of unitmass under a force exerted by a spring,which in view of (124) can be written in theformd 2x dxdt

'+C £iT +kx = 0, c> O. (138)

The reader will recall that in this equationc > 0 characterizes the drag of the mediumin which the object moves and k > 0characterizes the properties of the spring(the spring constant). The autonomous system corresponding to Eq. (138) has theformdx dy(ft=Y, (ft= -kx-cy. (139)

For this system the origin of the phase(x, y)-plane is the only singular point. Thekinetic energy of the object (of unit mass)is y2/2 and the potential energy (i.e, theenergy stored by the spring) isx

~ k6 d6 = +kx2•

o

This implies that the total energy of thesystem is

1 1V (x, y) === T y2+ T kx2. (140)

It is easy to see that V specified by (140)

Ch. 2. Qualitative Methods 1S9

is a positive definite function. And sincein the given caseav avaxX(x, Y)+ayY(x, y)

=kxy+y (-kx-cy) = -cy2~O,

this function is the Lyapunov function ofsystem (139), which means that the singular point 0 (0, 0) is stable.

In the above example the result wasobtained fairly quickly. This is not alwaysthe case, however. The formulated Lyapunov criterion is purely qualitative andthis does not provide a procedure forfinding the Lyapunov function even if weknow that such a function exists. Thismakes it much more difficult to determinewhether a concrete system is stable or not.

The reader must bear in mind that theabove criterion of Lyapunov must be seenas a device for finding effective indicationsof equilibrium. Many studies have beendevoted to this problem and a number ofinteresting results have been obtained inrecent years. *2.7 Simple States of Equilibrium

The dynamical interpretation of secondorder differential equations already implies

• The interested reader can refer to the book byE.A. Barbasbin, Lyapunov Functions (Moscow:Nauka, 1970) (in Russian).

tOO Differential Equations in Applications

that investigation of the nature of equilibrium states Of, which is the same, the singularpoints provides a key for establishing tilebehavior of integral curves.

It is also clear that, generally speaking,differential equations cannot be integratedin closed form. We need criteria that willenable us to determine the type of a singular point from the form of the initialdifferential equation. Unfortunately, asa rule it is extremely difficult to find suchcriteria, but it is possible to isolate certainclasses of differential equations for whichthis can be done fairly easily. Below weshow, using the example of an object of unitmass subjected to the action of linearsprings and moving in a medium withlinear drag, how some results of the qualitative theory of differential equations canbe used to this end. But first let us discussa system of the (122) type. It so happensthat the simplest case in establishing thetype of a singular point is when the Jacobianor functional determinant

ax ax---az ay

J (x, y) = ay ay

8Xayis nonzero at the point.

If (x*, y*) is a singular point ofsystem (122)and if J* = J (x*, Y*) =1= 0, then the typeof the singular point, which in the given

case is called a simple singular point,depends largely on the sign of constant J*.For instance, if J* is negative, the singularpoint (x*, y*) is a saddle point, and if J* ispositive, the singular point may be a vortex point, a nodal point, or a focal point.The singular point may be a vortex pointonly if the divergence

ax ayD(x y)=-+-, ax By

vanishes at the singular point, that is, onlyif D* = D (x*, y*) = O. Note, however,that the condition D* = 0 is generallyinsufficient for the singular point (x*, y*) tobe a vortex point. For a vortex point tobe present certain additional conditionsmust be met, conditions that include higher partial derivatives. And, generally,there can be an infinite number of suchconditions. But if functions X and Yarelinear in variables x and y, the conditionD* = 0 becomes sufficient for the singularpoint (x*, y*) to be a vortex point.

If J* > 0 but the singular point (x*, y*)is not a vortex point, a sufficiently smallneighborhood of this point is filled withtrajectories that either spiral into this pointor converge to it in certain directions. Here,if D* > 0, the singular point is reached ast -+ - 00 and is unstable while if D * < 0,the singular point is reached as t-o- +00and proves to be stable. If the phase tra-

(141)

f 92 Differential Equations in Applications

jectories that reach the singular point arespirals, we are dealing with a focal point,but if the integral curves converge to a singular point along a certain tangent, thepoint is a nodal point (Figure 62).

Irrespective of the sign of Jacobian J*,the tangents to the trajectories of the differential system (122) at a singular point (x*,y*) can be found from what is known as thecharacteristic equation

ay (x·, y*) '" BY (x·, y*) '"y ax x + iJy Y

'; = ax(x*, y*) rY ax (x*, y*) '" 'ax x+ ay Y

where

; = x - x*, y= y - y*. (142)

If X and Y contain linear terms, thepartial derivatives in Eq. (141) act as coefficients of x and y in the system obtainedfrom system (122) after introducing thesubstitution (142).

Equation (141) is homogeneous. Hence,

if we introduce the slope Iv = y;;of what isknown as exceptional directions, we havethe following quadratic equation for finding Iv:

X:A2 + (X: - yZ) 'A - y: = O. (143)

The discriminant of this equation is

~ = (X: + y:)2 - 4J* = D*2 - 4J*.

C:>

f..l

lj( ~

~~

.....:::.~

C:>

~

t~-..

tS'-

~ C\Jeo

(;,) bi>~

13- 0770

Hence, if J* < 0, which corresponds toa saddle point, Eq. (143) always gives tworeal exceptional directions. But if J* > 0,there are no real exceptional directions,or there are two, or there is one 2-folddirection. In the first case the singularpoint is either a vortex point or a focal point.

The existence of real exceptional directions means (provided that J* is positive)that there is a singular point of the nodaltype. For one thing, if there are two realexceptional directions, it can be provedthat there are exactly two trajectories (oneon each side) whose tangent at the singularpoint is one of the exceptional straight lines(directions) 'while all the other trajectories "enter" the singular point touching theother exceptional straight line (Figure 62a).

If ~ = 0 and Eq. (143) is not an identity,we have only one exceptional straight line.The pattern of the trajectories for this caseis illustrated by Figure 62b. It can beobtained from the previous case when thetwo exceptional directions coincide. Thesingular point divides the exceptionalstraight line into two half-lines, 11 and l2'while the neighborhood of the singular pointis divided into two sectors, one of which iscompletely filled with trajectories that"enter" the singular point and touch II andthe other is completely filled with trajectories that "enter" the singular point andtouch [2. The boundary between the sectors

ch. 2. Qualitative Methods i95

(145)

consists of two trajectories, one of whichtouches II at the singular point and the othertouches l2 at this point.

If in Eq. (143) all coefficients vanish, wearrive at an identity, and then all thestraight lines passing through the singularpoint are exceptional and there are exactlytwo trajectories (one on each side) thattouch each of these straight lines at thesingular point. This point (Figure 62c) issimilar to the point with one 2-fold realexceptional direction.

2.8 Motion of a Unit .. Mass ObjectUnder the Action of Linear Springsin a Medium with Linear Drag

As demonstrated earlier, the differentialequation describing the motion of a unitmass object under the action of linearsprings in a medium with linear drag hasthe formdi,x dxdt 2 +cdt+kx=O. (144)

So as not to restrict the differential model(144) to particular cases we will not fix thedirections in which the forces -c (dx/dt)and -kx act. As shown earlier, withEq. (144) we can associate an autonomoussystem of the form

dx dy kdt=Y, d't= - x-eYe

13*

f96 Differential Equations in Applications

If we now exclude the trivial case withk = 0, which we assume is true for themeantime, the differential system (145) hasan isolated singular point at the origin.System (145) is a particular case of thegeneral system (122). In our concrete example

X (z, y) = y, y (x, y) = -kx - cy,

the Jacobian J (x, y) = k, and the divergenceD (z, y) == -c. The characteristic equation assumes the form

A,2 +CA+k= 0,where t1 == c2 - 4k is the discriminant ofthis equation. In accordance with the resultsobtained in Section 2.7 we arrive at thefollowing cases.

(1) If k is negative, the singular point isa saddle point with one positive and onenegative exceptional direction. The phasetrajectory pattern is illustrated in Figure 63, where we can distinguish betweenthree different types of motion. When theinitial conditions correspond to point a,at which the velocity vector is directed tothe origin and the velocity is sufficientlygreat, the representative point moves alonga trajectory toward the singular point ata decreasing speed; after passing the originthe representative point moves away fromit at an increasing speed. If the initialvelocity decreases to a critical value, which

Ch, 2. Qualitative Methods

Fig. 63

x

197

corresponds to point b, the representativepoint approaches the singular point ata decreasing speed and "reaches" the originin an infinitely long time interval. Finally,if the initial velocity is lower than thecritical value and corresponds, say, to pointC, the representative point approaches theorigin at a decreasing speed, which vanishesat a certain distance Xl from the origin.At point (Xl' 0) the velocity vector reversesits direction and the representative pointmoves away from the origin.

If the phase point corresponding to theinitial state of the dynamical system liesin either one of the other three quadrants,the interpretation of the motion is obvious.


Fig. 64

Fig. 65

(2) If k > 0, then J* is positive and thetype of singular point depends on thevalue of c. This leaves us with the following possibilities:

(2a) If c = 0, that is, drag is nil, thesingular point is a vortex point (Figure 64).The movements are periodic and theiramplitude depends on the initial conditions.

Ch, 2. Qualitative Methods 199

(2b) If c > 0, that is, damping is positive, the divergence D == ax /a..~ + fJYloy isnegative and, hence, the representativepoint moves along a trajectory toward theorigin and reaches it in an infinitely longtime interval.

More precisely:(2bJ ) If ~ < 0, that is, c~ < 4k, the Sill-

gular point proves to he a focal point(Figure 65) and, hence, the dynamicalsystem performs damped oscillations aboutthe state of equilibriurn with a decayingamplitude.

(2b 2) If !1 = 0, that is, c2 = 4k, thesingular point is a nodal point with a singlenegative exceptional direction (Fignre 66).The motion in this case is aperiodic and corresponds to the so-called critical damping.

(2b:l) If ~ > 0, that is, c~ > 4k, thesingular point is a nodal point with t\VO

negative exceptional directions (Figure 67).Qualitatively the motion of the dynamicalsystem is the same as in the previous caseand corresponds to damped oscillations.

From the above results it follows thatwhen c >0 and k >0, that is, drag is positive and the restoring force is attractive,the dynamical system tends to a state ofequi librium and its motion is sta hle.

(2c) If c < 0, that is, damping is negative" the qualitative pattern of the phasetrajectories is the same as in the case (2b),the only difference being that here the


Fig. 66

Fig. 67


Fig. 68

c

Fig. 69

dynamical system ceases to be stable.Figure 68 contains all the above results

and the dependence of the type of singular


point on the values of parameters c and k.Note that the diagrams can also he interpreted as a summary of the results of studiesof the types of singular points of system (122)when J* * 0 at c = -D* and k = J*.However, the fact that c = 0 does notgenerally mean that system (122) possessesa vortex point, and the fact that k = 0 doesnot mean that the system of a general type11a8 no singular point. These cases belongto those of complex singular points, whichwe consider below.

Returning to the dynamical system considered in this section, we note that if k = 0(c =F 0), the autonomous system (145) corresponding to Eq. (144) assumes the form

dx dy£it=Y, d"t=-cy.

This implies that the straight line y = 0 isdensely populated by singular points, withthe phase trajectory pattern shown inFigure 69.

Finally, if k == c = 0, Eq. (144) assumesthe form

The respective autonomous system is

dx dy(fj=Y' CU==O.


Here, as in the previous case, the .1~ axis isdensely populated by singular points. Thorespective phase trajectory pattern isshown in Figure 70.

2.9 Adiabatic Flowof a Perfect Gas Through a Noz.sleof Varying Cross Section

A study of the fJow of compressible viscousmedia is highly important from the practical viewpoint. For one thing, such flowemerges in the vicinity of a wing and fuselage of an airplane; it also influences theoperation of steam and gas turbines, jetengines, the nuclear reactors.

Below we discuss the flow of. a perfectgas through a nozzle with a varying crosssection (Figure 7·1); the specific heat capacity of the gas is c p and the nozsle's varyingcross-sectional area is denoted by A. Thoflow is interpreted as one-dimensional, thatis, all its properties are assumed to be uniform in a single cross section of the nozzle.Friction in the boundary layer is causedby the tangential stress 't given by the formula

T = qpv2/2, (146)

where q is the friction coefficient dependingbasically on the Reynolds number butassumed constant along the nozzle. p the


Fig. 70..

Fig. 71

flux density, and v the flow velocity. Finally, we assume that adiabaticity conditionsare met, that is, drag, combustion, chemical transformations, evaporation, and condensation are excluded.

One of the basic equations describingthis type of flow is the well-known conti-


nuity equation, which in the given caseis written in the form

w == pAv, (147)

where the flux variation rate w is assumedconstant. From this equation it followsthat

~-+- dA +~ ==0.P I A v

(148)

Let us now turn to the equation for theenergy of steady-state flow. We note thatgenerally such an equation links the external work done on the system and the actionof external heat sources with the increasein enthalpy (heat content) in the flow andthe kinetic and potential energies. In ourcase the flow is adiabatic; hence, the energybalance equation can be written in the form

o ==: W (h + dh) - wh+ w [v2/2 + d (v2/2)1 - wv2/2,

or

dh + d (v2/2) == 0, (149)

where h is the enthalpy of the flow (thethermodynamic potential) at absolute temperature T. But in Eq. (149) dh = cpdTand, therefore, we can write the equationfor the flow energy as

c]JdT + d (v2/2) == O. (150)


Now let us derive the momentum equation for the flow. Note that here the commonapproach to problems involving a steadystate flow is to use Newton's second law.Assuming that the divergence angle of thenozzle wall is small, we can write themomentum equation in the form

pA + p dA - (p + dp) (A + dA) - 1" dA= w dv,

or

-A dp - dA dp - T rlA === w dv, (151)

where p is the static pressure.The term dA dp in Eq. (151) is of a

higher order than the other terms and,therefore, we can always assume that themomentum equation for the flow has theform

-A dp - 't dA = to du. (152)

If we denote by D the hydraulic diameter,we note that its variation along the nozzle'saxis is determined by a function F such thatD = F (x), where x is the coordinate alongthe nozzle's axis. From the definition of thehydraulic diameter it follows that

(153)

Noting that pv2/2 = 1'pM2, where "( is thespecific heat ratio of the medium, and M


is the Mach number, we can write formula(146) thus:

't == q)'pM2. (154)

Combining this with Eqs. (147) and (153),we arrive at the following representation for the momentum equation (152):

~+l'M2(4q~+ dv2 )=0. (155)

p D VI

Denoting the square of the Mach number byy, employing Eqs. (148), (150), and (155),and performing the necessary algebraictransformations, we arrive at the followingdifferential equation:

d 4y ( 1+ 1';1 y) (yqy-F' (x»

d; = (1-y)F(x) (156)

where the prime stands for derivative ofthe respective quantity.*

The denominator of the right-hand side ofEq. (156) vanishes at y = 1, that is, whenthe Mach number becomes equal to unity.This means that the integral curves of thelast equation intersect what is known as thesonic line and have vertical tangents at theintersection points. Since the right-hand

• See J. Kestin and S.K. Zaremba, "One-dimensional high-speed flows. Flow patterns derivedfor the flow of gases through nozzles, includingcompressibility and viscosity effects," A ircrajtEngin. 25, No. 292: 172-175, 179 (1953).


side of Eq. (156) reverses its sign in theprocess of intersection, the integral curves"flip over" and the possibility of inflectionpoints is excluded. The physical meaningof this phenomenon implies that alongintegral curves the value of x must increasecontinuously. Hence, the section on whichthe integral curves intersect the sonic linewith vertical tangents must be the exitsection of the nozzle. Thus, the transitionfrom subsonic flow to supersonic (and back)can occur inside the nozzle only througha singular point with real exceptional directions, that is, through a saddle point ora nodal point.

The coordinates of the singular points ofEq. (1!16) are specified by the equations

y* == 1, F'(x*) == yq,

which imply that these points are situatedin the diverging part of the nozzle. A saddlepoint appears if J* is negative, that is,F" (x*) > O. Since q is a sufficiently smallconstant, a saddle point appears near thethroat of the nozzle. A nodal point, on theother hand, appears only if F" (x*) is negative. 'rhus, a nodal point emerges in thepart of the nozzle that lies behind an inflection point of the nozzle's profile or, inpractical terms, at a certain distance fromthe throat of the nozzle, provided that theprofile contains an inflection point.


From the characteristic equation

F(x*) "A2 + 2qy (y + 1) "A

- 2 (1' + 1) F" (x*) == 0

we can see that the slopes of the two exceptional directions are opposite in sign in thecase of a saddle point and have the samesign (are negative) in the case of a nodalpoint. This means that only a saddle pointallows for a transition from supersonic tosubsonic velocities and from subsonic tosupersonic velocities (Figure 72). The caseof a nodal point (Figure 73) allows for acontinuous transition only from supersonicto subsonic flow.

Since Eq. (156) cannot be integrated inclosed form, we must employ numericalmethods of integration in any further discussion. It is advisable in this connectionto begin the construction of the four separatrices of a saddle point as integral curvesby allowing for the fact that the singularpoint itself is a point, so to say, from whichthese integral curves emerge. Such a construction is indeed possible since the characteristic equation provides us with thedirection of the two tangents at the singularpoint S (x*, 1). If this is ignored and themotion monitored as beginning at pointsa and b in Figure 72, which lie on differentsides of a separatrix, the correspondingpoints move along curves (X and ~ thatstrongly diverge and, hence, provide no in-

1'-0770


,..-----r--+--.....:MJ~----ip......--+---

x*

Fig. 72

J~----.;~-----

o

Fig. 73

formation about the integral curve (theseparatrix) that "enters" point S. On the


other hand, if we move along an integralcurve that "emerges" from point Sand weassume that the initial segment of the curvecoincides with a segment of an exceptionalstraight line, the error may be minim izedif we allow for the convergence of integralcurves in the direction in which the valuesof x diminish.

Figure 72 illustrates the pattern of integral curves in the vicinity of a singularpoint. The straight line passing throughpoint Xt (the throat of the nozzle) corresponds to values at which the numeratorin the right-hand side of Eq. (156) vanishes,which points to the presence of extrema.

2.10 Higher-Order Pointsof Equilibrium

In previous sections we studied the typesof singular points that emerge when theJacobian J* is nonzero. But suppose thatall the partial derivatives of the functions Xand Y in the right-hand sides of system (122)vanish up to the nth order inclusive. Thenin the vicinity of a singular point theremay be an infinitude of phase-trajectorypatterns. However, if we exclude the possibility of equilibrium points of the vortexand focal types emerging in this picture,then it appears that the neighborhood ofa singular point can be broken down into

a finite number of sectors belonging tothree standard types. These are the hyperbolic, parabolic, and elliptic. Below we describe these sectors in detail, but first let usmake several assumptions to simplify matters.

We assume that the origin is shifted tothe singular point, that is, x* = y* = 0;the right-hand sides of system (122) canbe written in the form

x (x, y) = X n (x, y) + <D (x, y),y (x, y) = Y n (x, y) + 'I' (x, y), (157)

where X nand Y n are polynom ials of degree n homogeneous in variables x and y(one of these polynomials may be identicallyzero), and the functions and '¥ have inthe neighborhood of the origin continuousfirst partial derivatives. In addition, weassume that the functions

<D (I, y) <Dx (I, y) <l>y (I, y)(x 2 + y 2)(n +l )/2 ' (x 2 + y2)n/ 2 ' (x 2+ y2)n/2 '

'I' (I, y) 'I'x (I, y) 'I'y (x, y)(x2+y2)(n+l)/2' (x2+ y2)n/'1 ' (x2+ y2)n/2

are bounded in the neighborhood of theorigin. Under these assumptions the following assertions hold true.

(1) Every trajectory of the system ofequations (122) with right-hand sides of the(157) form that "enters" the origin alonga certain tangent touches one of the ex-


ceptional straight lines specified by theequation

xY n (x, y) - yX n (x, y) = O. (158)

Since the functions X nand Y n are homogeneous, we can rewrite Eq. (158) as anequation for the slope A == y/x. Then theexceptional straight line is said to be singular if

X n (x, y) = Y n (x, y) = 0

on this straight line. Some examples ofsuch straight lines are shown in Figure 62.The straight lines defined by Eq. (158) butnot singular are said to be regular.

(2) The pattern of the phase trajectoriesof (122) in the vicinity of one of the tworays that "emerge" from the origin andtogether form an exceptional straight linecan be studied by considering a small disk(centered at the origin) from which weselect a sector limited by two radii lyingsufficiently close to the rayon both sidesof it. Such a sector is commonly known asa standard domain.

More than that, in the case of a regularexceptional straight line, which corresponds to a linear factor in Eq. (158), thestandard domain considered in a disk ofa sufficiently small radius belongs to eitherone of two types: attractive or repulsive.

(2a) The attractive standard domain ischaracterized by the fact that each tra-


Fig. 74

Fig. 75

jectory passing through it reaches theorigin along the tangent that coincideswith the exceptional straight line (Figure 74).

(2b) The repulsive standard domain ischaracterized by the fact that only onephase trajectory passing through it reachesthe singular point along the tangent thatcoincides with the exceptional straight line.All other phase trajectories of (122) thatenter the standard domain through theboundary of the disk leave the disk bycrossing one of the radii that limit thedomain (Figure 75).

ChI 2. Qualitative Methods 215

Let us turn our attention to the followingfact. If the disk centered at the origin issmall enough, the two types of standarddomains can be classified according to thebehavior of the vector (X, Y) on theboundary of the domain. The behavior ofvector (X, Y) can be identified here withthe behavior of vector (X n' Y n). Morethan that, it can be demonstrated that if,as assumed, a fixed exceptional directiondoes not correspond to a multiple root ofthe characteristic equation, the vector considered on one of the radii that limit thedomain is directed either inward or outward. Then if in the first case the vectorconsidered on the part of the boundaryof the standard domain that is the arc ofthe circle is also directed inward, and inthe second case outward, the standarddomain is attractive. But if the oppositesituation is true, the standard domain isrepulsive. It must be noted that in any casethe vector considered on the part of theboundary that is the arc of the circle isalways directed either inward or outwardsince it is almost parallel to the radius.

Standard domains corresponding to singular exceptional directions or multipleroots of the characteristic equation havea more complicated nature, but since tosome extent they constitute a highly rarephenomenon, we will not describe them here.

If we now turn, for example, to a saddle


point, we note that it allows for fourrepulsive standard domains. In the neighborhood of a singular point of the nodaltype there are two attractive standarddomains and two repulsive.

(3) If there exist real-valued exceptionaldirection straight lines, the neighborhoodof a singular point can be divided into afinite number of sectors each of which isbounded by the two phase trajectories of(122) that "enter" the origin along definitetangents. Each of such sectors belongs toone of the following three types.

(3a) The elliptic sector (Figure 76) contains an infinitude of phase trajectoriesin the form of loops passing through theorigin and touching on each side of theboundary of the sector.

(3b) The parabolic sector (Figure 77)is filled with phase trajectories that connectthe singular point with the boundary of theneighborhood.

(3c) The hyperbolic sector (Figure 78)is filled with phase trajectories that approach the boundary of the neighborhoodin both directions.

More precisely:(4a) Elliptic sectors are formed between

two phase trajectories belonging to twosuccessive standard domains, both of whichare attractive.

(4b) Parabolic sectors are formed betweentwo phase trajectories belonging to" two


s

Fig. 76 Fig. 77

s

Fig. 78

successive standard domains one of whichis attractive and the other repulsive. Allphase trajectories that pass through thelatter domain touch at the singular pointof the exceptional straight line that definesthe attractive domain.


(4c) Hyperbolic sectors are formed between two phase trajectories belonging totwo successive repulsive standard domains.

For example, it is easy to distinguish thefour hyperbolic sectors at a saddle pointand the four parabolic sectors at a nodalpoint. Elliptic sectors do not appear inthe case of simple singular points, wherethe Jacobian J* is nonzero.

If a singular point does not allow for theexistence of real exceptional directions,the phase trajectories in its neighborhoodalways possess the vortex or focal structure.*

2. t t Inversion with Respectto 8 Circle and HomogeneousCoordinates

Above we described methods for establishing the local behavior of phase trajectoriesof differential systems of the (122) type inthe neighborhood of singular points. Andalthough in many cases all required information can be extracted by following thesemethods, there may be a need to study the

• Methods that make it possible to distinguishbetween a vortex point and a focal point are discussed, for example, in the book by V. V Arnel'kin, N.A. Lukashevich, and A.P. Sadovskii,N onlinear Vibrations in Second-Order Systems(Minsk: Belorussian Univ. Press, 1982) (in Russian).


trajectory behavior in infinitely distantparts of the phase plane, as x2 + y2 ~ 00.

A simple way of studying the asymptoticbehavior of the phase trajectories of thedifferential system (122) is to introducethe point at infinity by transforming theinitial differential system in an appropriatemanner, say by inversion, which is definedby the following Iormulas:

x- 6 y __ 11- S2+ TJ 2 ' - S2+1l 2

(6= X2~y2' 11= X 2t y2 ). (159)

Geometrically this transformation constitutes what has become known as inversionwith respect to a circle and maps the origininto the point at infinity and vice versa.Transformation (159) maps every finitepoint M (x, y) of the phase plane into pointM' (~, ",,) of the same plane, with points Mand M' lying on a single ray that emergesfrom the origin and obeying the conditionOM X OM' = r2 (Figure 79). It is wellknown that such a transformation mapscircles into circles (straight lines are considered circles passing through the pointat infinity). For one thing, straight linespassing through the origin are invariantunder transformation (159). Hence, theslopes of asymptotic directions are theslopes of tangents at the new origin ~ =11 = o. Note that in the majority of


Fig. 79

N

Fig. 80

cases the new origin serves as a singularpoint. The reasons for this are discussedbelow.


As for the question of how to constructin the old (x, y)-plane a curve that has adefinite asymptotic direction, that is, hasa definite tangent at the origin of the new(~, 11 )-plane, this can be started by considering the (~, "l)-plane, more precisely,by considering this curve in, say, a unitcircle in the (~, "1)-plane. The fact is thatsince a unit circle is mapped, via transformation (159), into itself, we can alwaysestablish the point where the curve intersects therespective unit circle in the (x, y)-plane;any further investigations can be carriedout in the usual manner.

We also note that completion of the(x, y)-plane with the point at infinity istopologically equivalent to inversion ofthe stereographic projection (Figure 80),in which the points on a sphere are mappedonto a plane that is tangent to the sphereat point S. The projection center N is theantipodal point of S. It is clear that theprojection center N corresponds to thepoint at infinity in the (x, y)-plane. Conversely, if we map the plane onto thesphere, a vector field on the plane transforms into a vector field on the sphere andthe point at infinity may prove to be a singular point on the sphere.

Although inversion with respect to acircle is useful, it proves cumbersome andinconvenient when the point at infinity hasa complicated structure. In such cases


another, more convenient, transformation ofthe (x, y)-plane is used by introducinghomogeneous coordinates:

x == ~/z, Y == ll/Z.

Under this transformation, each point ofthe (x, y)-plane is associated with a tripleof real numbers (S, 11, z) that are notsimultaneously zero and no difference ismade between the triples (6, 11, s) and(k~, k11, kz) for every real k =1= O. If apoint (x, y) is not at infinity, z =1= O. Butif z = 0, we have a straight line at infinity.The (x, y)-plane completed with the straightline at infinity is called the projective plane.Such a straight line may carry several singular points, and the nature of these pointsis usually simpler than that of a singularpoint introduced by inversion with respectto a circle.

If we now consider a pencil of lines anddescribe its center with a sphere of, say,unit radius, then each line of the pencilintersects the sphere at antipodal points.This implies that every point of the" projective plane is mapped continuously andin a one-to-one manner onto a pair of antipodal points on the unit sphere. Thus, theprojective plane may be interpreted as theset of all pairs of antipodal points on a unitsphere. To visualize the projective plane,we need only consider one-half of the sphereand assume its points to be the points of the


Fig. 81

projective plane. If we orthogonally projectthis hemisphere (say, the lower one) ontothe a-plane, which touches it at pole S(Figure 81), the projective plane is mappedonto a unit disk whose antipodal pointson the boundary are assumed identical.Each pair of the antipodal points of theboundary corresponds to a line at infinity,and the completion of the Euclidean planewith this line transforms the plane intoa closed surface, the projective plane.

2.12 Flow ofa Perfect GasThrough a Rotating Tubeof Uniform Cross Section

In some types of turboprop helicopters andairplanes and in jet turbines, the gaseousfuel-air mixture is forced through rotating


r

c

e

Fig. 82

tubes of uniform cross section installedin the compressor blades and linked by ahollow vertical axle. To establish the optimal conditions for rotation we must analyzethe flow of the mixture through a rotatingtube and link the solution with boundaryconditions determined by the tube design.In a blade the gaseous mixture participatesin a rotational motion with respect to theaxis with constant angular velocity roand moves relative to the tube with anacceleration v (dv/dr), where v is the speedof a gas particle with respect to the tube,and r is the coordinate measured along therotating compressor blades.

Figure 82 depicts schematically a singlerotating tube of a compressor blade.* It is

• See I. Kestin and S.K. Zaremba, "Adiabaticone-dimensional flow of a perfect gas through arotating tube of uniform cross section," Aeronaut.Quart. 4: 373-399 (1954).


assumed that the fuel-air mixture, whoseinitial state is known, is supplied alongthe hollow axle to a cavity on the axle inwhich the flow velocity may be assumedinsignificant. The boundary of the cavityoccupied by the gas is denoted by a, thegas is assumed perfect with a specific heatratio 'V, and all the processes that the gasmixture undergoes are assumed reversiblyadiabatic (exceptions are noted below).

It is assumed that the gas expands througha nozzle with an outlet cross section b thatat the same time is the inlet of a tube ofuniform cross section. The expansion ofthe gas mixture from state a to state bis assumed to proceed isentropically; wedenote the velocity of the gas after expansion by VI and the distance from the rotationaxis 0 1 to the cross section b by r i .

When passing through the tube, whoseuniform cross-sectional area will be denotedby A and whose hydraulic diameter by D,the gas is accelerated thanks to the combined action of the pressure drop and dynamical acceleration in the rotation compressor blade. We ignore here the effect producedby pressure variations in the tube (if theyexist at all) and by variations in pressuredrop acting on the cross section plane, bothof which are the result of the Coriolis force.The last assumption, generally speaking,requires experimental verification, sincethe existence of a lateral pressure drop may15-0770


serve as a cause of secondary flows. But ifthe diameter of the tube is small comparedto the tube's length, such an assumptionis justified.

It is now clear that the equations of momentum and energy balance for a compressible mixture traveling along a tube ofuniform cross section must be modified soas to allow for forces of inertia that appearin a rotating reference frame. As for thecontinuity equation, it remains the same.

Now let us suppose that starting fromcross section c at the right end of the tubethe gas is compressed isentropically andpasses through a diverging nozzle. In theprocess it transfers into a state of rest withrespect to the compressor blade in the secondcavity at a distance '3 from the rotationaxis and reaches a state with pressure Pd

and temperature Td-

From the second cavity the gaseous mixture expands isentropically into a converging or converging-diverging nozzle in sucha manner that it leaves the cavity at rightangles to the tube's axis. This producesa thrust force caused by the presence ofa torque.

Below for the sake of simplicity we assume that the exit nozzle is a convergingone and has an exit (throat) of area A *.Denoting the external (atmospheric) pressure by P a' we consider two modes of passage of the mixture through the nozzle.

The first applies, to a situation in whichthe Pa-to-Pd ratio exceeds the criticalvalue, or

Pa!Pd > (2/("1 + 1»,,/(Y-1).

In this case the flow at. the nozzle's throatis subsonic and, hence, pressure P 3 at thethroat is equal to the atmospheric pressure,that is, P 3 == Pa. The second case appliesto a situation in which the Pa-to-Pd. ratiois below the critical value, or

Pa/P d < (2/("1 + 1)"/(V-1).

The pressure Ps at the nozzle's throat hasa fixed value that depends on Pd but noton P a. Hence,P a .= (2/("1 + 1»)"/(1'-1) Pd -

In the latter case the flow at the nozzle'sthroat has the speed of sound

va = (2/(y + 1»1/2 ad,

where ad depends only on temperature T d.

In further analysis the flow is assumedadiabatic everywhere and isentropic everywhere except in the tube between cross sections band c.

As in the -case where we derive the differential equation that describes the adiabatic flow of a perfect gas through a nozzleof varying cross section, let us now considerthe continuity equation, the equation of

15*


p

r s dr

o s d»

Fig. 83

momentum, and the equation of energybalance. For instance, the equation of continuity in this case has the form

v Vs A*1J'==-y==v;- A=const, (160)

with V the specific volume and 'I' the fluxdensity, or

¢ = m'lA,

where m' is the flux mass.To derive the equation of motion, or the

momentum equation, we turn to Figure 83.We note that the dynamical effect of therotational motion of the compressor bladecan be used to describe the flow with respect to the moving tube, where in accordance with the D'Alembert principle theforce of inertia

dI =~w2rdrV


is assumed to act in the positive directionof r. Hence, the element of mass dm =(A/V) dr moves with an accelerationv (dv/dr) caused by the combined actionof the force of inertia dI, the force of pressure A dP, and the friction force dF =(4A/D) dr. Here i = Iv (v2 /ZV), where Ivdepends on the Reynolds number R. Asa first approximation we can assume thatIv remains constant along the entire tube.With this in mind, we can write the momentum equation in the formA dv 2AAvz A dV dr v dr = - A dP - VD dr +V w2r r,

or after certain manipulations,2AV dP +v du-l- D v2dr - w2r dr = o. (161)

As for the energy-balance equation, it issimple to derive if we use the first law ofthermodynamics for open systems and bearin mind that the amount of work performedby the system is w2r dr. Thus

dh + v dv - w2r dr = 0,

where h is the enthalpy. If we define thespeed of sound a via the equation

h=~1'-1 '

we find that

ada+ '\';1 vdv- '\';1 w2rdr=O.

(164)

(165)

(166)


This leads us to the following energy-balanceequation:

2 2 l' - 1 2 '\' - 1 2 2 162)ao ==a -t- -2- v - -2- w r · (

If we now introduce the dimensionlessquantitieslYlo = vlao' x = riD, G2 = w2D2Ia~, (163)we arrive at the equation

~-1- 1'-1 M2+ '\'-1 G2 2a2 - 2 0 2 x ·

By excluding the pressure and specificvolume from the basic equations (160),(161), and (164) we can derive an equationthat links the dimensionless quantity M 0

with the dimensionless distance. This equation serves as the basic equation for solvingour problem. The continuity equation (160)implies that

V == vl¢.Then, combining

"'Pva2 = yPV = _r_1P

with Eq. (162), we arrive at the followingrelationships:

(a~ l' - 1 'V - 1 w2

)p== ---- t: ~1--- -- r2 Ill,'Vv 21' 2)' v 't ,

dP _ 1'-1 w'!.r ('V-1 + a~"""dr - -V- -v- -- \~ yv 2

+ 1'-1 W 2r2) dv2y-V2 """dr-

ChI 2. Qualitative Methods 231

Substituting the value of V from Eq. (165)and the value of dP/dr from Eq. (166) intothe momentum equation (161) and allowingfor the relationship (163), we arrive at thedifferential equation

dMg = 2M6 (2A1'M~-G2x) • (167)dx 1- ,\,+1 M:+ '\'-1 G2x2

2 2

Assuming that

M~= y, m= 2'Ay,

1q= 2 (1'-1),

we can rewrite Eq. (167) as

dy 2y (my-G2x)dz 1-py+ qG2x2 •

(168)

The differential equation (168) is theobject of our further investigation. Introducing the variables (159), we reduce it to

dn _ 2~1l (:S2 - P1l8+ qG2~2)+21] (l)2_~2) Qdf- A (g2-pT)E+qG2~2)+4~T)2Q '

(169)

where 3 == £2 + 112, Q ==mll- G2~, A == £2 - t]2.Clearly, the origin constitutes a singularpoint for Eq. (169). The lowest-order termsin both numerator and denominator arequadratic. If we discard higher-order terms,


as we did on p. 211 when discussing higherorder equilibrium points, we find that

y 4 (6, ,,)== 2qG2~311 + 211 (1l2 - ~2) (mll - G2~),

X 4 (6, ,,)== qG2 S2 (~2 - 112) + 4~112 (mll - G2~),

and, hence, the origin is a higher-ordersingular point.

The characteristic equation of the differential equation (169) assumes the followingform after certain manipulations:

611 (62 + 112) [(q + 2) G2S- 2mrt] = O.(170)

This leads us to three real-valued exceptional straight lines:

(a) S = 0, (b) tl == 0,(c) (q + 2) G2S- 2mtl == O. (171)

Each is a regular straight line and corresponds to one of the factors in Eq. (170).

It is easy to see that for sand 11 posi ti vethe value of X 4 (~, 11) is positive in theneighborhood of all three exceptionalstraight lines, with the result that theexpression

Y4 (;, 11) _~X 4 (~, ,,) s

~T) (~2+ 112) [(q + 2) G2~- 2m'll== £ [qG262 (S2- '12)+4~t)2 (mt]- G2S)) (172)


has the same sign in both the first quadrantand the neighborhood of the straight linesas the left-hand side of (171c) and, hence,is negative below the straight line (171c)and positive above. The explanation liesin the fact that (172) may change sign onlywhen an exceptional straight line is crossed.Geometrically this means that the righthand side of the differential equation

determining the behavior of integral curvesin a disk of sufficiently small radius andcentered at the origin fixes an angle greaterthan the angle of inclination of the radiithat lie in the first quadrant between thestraight lines (171h) and (171c). As forthe radii lying between the straight lines(171c) and (171a), on them the right-handside of this differential equation fixes anangle that is smaller than the angle ofinclination of these radii (Figure 84). Thus,the standard domain containing the straightline (171c) must be repulsive, while thedomains containing the straight lines (171a)and (171h) must be attractive. In view ofthe symmetry of the field specified by vector(X 4 , Y4)' the above facts remain valid forstandard domains obtained from those mentioned earlier by a rotation about thesingular point through an angle of 180°.


Fig. 84

Thus, there are exactly two integralcurves that "enter" the singular point alongthe tangent (171c) and an infinitude ofintegral curves that touch the coordinateaxes (171a) and (171h) at the point of rest.

We see then that the second and fourthquadrants contain elliptic sectors sincethey lie between two successive attractivestandard domains (Figure 85). Each of thefirst and third quadrants is divided intotwo sectors by the integral curves thattouch the exceptional straight line (171c)at the origin. These sectors are parabolicbecause they lie between two successivestandard domains, one of which is attractiveand the other repulsive. More than that,all the integral curves except those thattouch the straight line (171c) touch the


Fig. 85

235

coordinate axes (171a) and (171h) at thesingular point.

Basing our reasoning on physical considerations, we can analyze the differentialequation (168) solely in the first quadrantof the (x, y)-plane. Turning to this plane,we see that there exists exactly one integralcurve having an asymptotic direction witha slope (q + 2) G2/2m. All other integralcurves allow for the asymptotic directionof one of the coordinate axes. Indeed, it iseasy to prove that all these integral curvesasymptotically approach one of the coordinate axes, that is, along each of them inthe movement toward infinity not only doesy/x -* 0 or x/y -+ 0 but so does y --+ 0 orx ~ 0 (with x -+ 00 or y -+ 00, respec-


o-------------~...a:

Fig. 86

tively). This is illustrated in Figure 86,where sume integral curves have been plotted at great values of x and y. Note that thepattern of integral curves at a finite distance from the origin depends primarilyon the position of the singular points, and.a special investigation is required to clarifyit. It can also be proved that the integralcurves that approach the coordinate axesasymptotically (Figure 86) leave the firstquadrant at a finite distance from theorigin.

2. 13 Isolated Closed Trajectories

We already know that in the case of asingular point of the vortex type a certainregion of the phase plane is completelyfilled with closed trajectories. However,


a more complicated situation may occurwhen there is an isolated closed trajectory,that is, a trajectory in a certain neighborhood of which there are no other closedtrajectories. This case is directly linkedwith the existence of isolated periodicsolutions. Interestingly, only nonlinear differential equations and systems can haveisolated closed trajectories.

Isolated periodic solutions correspondto a broad spectrum of phenomena and processes occurring in biology, radiophysics,oscillation theory, astronomy, medicine,and the theory of device design. Such solutions emerge in differential models in economics, in various aspects of automaticcontrol, in airplane design, and in otherfields. Below we study the possibility ofisolated periodic solutions emerging in processes that occur in electric circuits; we alsoconsider as a model the nonlinear differential system

:: = -y + x (1 - x2 _ y2),

:; = x + y (1 - x2 _ y2).(173)

To solve this system, we introduce polarcoordinates rand Ot where x = r cos eand y = r sin 8. Then, differentiating therelationships x2 + y2 = r2 and B =


tan -1 (y/X) with respect to t, we arrive at

dx -L dy __ dr dy dx _ -2 dOx{ft I Ydt-r{ft, xdt- Y dT - 1 {ft-

(174)

Multiplying the first equation into x andthe second into y, adding the products,and allowing for the first relationship in(t74), we find that

drr (ft== r2 (1- r2) . (175)

Multiplying the second equation in (t 73)into x and the first into y, subtracting oneproduct from another, and allowing for thesecond relationship in (174) , we find that

dOr 2 -- r2

dt - • (176)

System (173) has only one singular point,o (0, 0). Since at the moment we are onlyinterested in constructing trajectories, wecan assume that r is positive. Then Eqs.(175) and (176) imply that system (173)can be reduced to the form

dr _ (1 2) de - 1dT- r -r, (ft- · (177)

Each of these equations can easily be integrated and the entire family of solutions,


as can easily be seen, is given by the formulas

1r = , e~ t + to' (178)

V1+Ce- 2t

or, in terms of the old variables x and y,

cos (t + to) sin (t+ to)x= , y=--;===

1/1+ ce-» -V1+ Ce- 2t

If now in the first equation in (178) we putC = 0, we get r = 1 and e = t + toThese two relationships define a closedtrajectory, a circle x2 + y2 ' 1. If C isnegative, it is clear that r is greater thanunity and tends to unity as t -+- +00. Butif C is positive, it is clear that r is less thanunity and tends to unity as t -+ +00. Thismeans that there exists only one closedtrajectory r == .1 which all other trajectoriesapproach along spirals with the passage oftime (Figure 87).

Closed phase trajectories possessing suchproperties are known as limit cycles or,more precisely, (orbitally) stable limit cycles.In fact, there can be two additional typesof limit cycles. A limit cycle is said to be(orbitally) unstable if all neighboring trajectories spiral away from it as t ~ +00.And a limit cycle is said to be (orbitally)half-stable if all neighboring trajectorieson one side (say, from the inner side) spiralon to it and all neighboring trajectories


Fig. 87

on the other side (say, on the outer side)spiral away from it as t -+ +00.

In the above example we were able tofind in explicit form the equation of a closedphase trajectory, but generally, of course,this cannot be done. Hence the importancein the theory of ordinary differential equations of criteria that enable at least specifying the regions where a limit cyclemay occur. Note that a closed trajectory of(122), if such a trajectory exists, containswithin its interior at least one singularpoint of the system. This, for one thing,implies that if there are no singular pointsof a differential system within a region ofthe phase plane, there are no closed trajectories in the region either.


Fig. 88

Let D be a bounded domain that liestogether with its boundary in the phaseplane and does not contain any singularpoints of system (122). Then the PoincareBendixson criterion holds true, namely, ifr is a trajectory of (122) that at the initialmoment t = to emerges front a point thatlies in D and remains in D for all t ~ to'then r is either closed or approaches a closedtrajectory along a spiral with the passage oftime.

We illustrate this in Figure 88. Here Dconsists of two closed curves r 1 and r 2

and the circular domain between them.With each boundary point of D we associatea vector

V (x, y) = X (x, y) i + Y (x, y) j.

Then, if a trajectory r that emerges at theinitial moment t = to from a boundarypoint, enters D, and remains there at all

t 6-0770


moments t ~ to, then, according to theabove criterion, it will along; a spiralapproach a closed trajectory r 0 that liesentirely in D. The curve r 0 must surrounda singular point of the differential system,point P, not lying in D.

The differential system (173) providesa simple example illustrating the application of the above criterion in Iinding limitcycles. Indeed, system (173) has only onesingular point, 0 (0, 0), and, therefore,the domain D lying between the circleswith radii r = 1/2 and r = 2 contains nosingular points. The first equation in (177)implies that dr/dt is positive on the innercircle and negative on the outer. Vector V,which is associated with the points on theboundary of D, is always directed into D.This means that the circular domain lyingbetween circles with radii r = 1/2 andr == 2 must contain a closed trajectory uf thedifferential system (173). Such a closedtrajectory docs indeed exist, it is a circle ofradius r = 1.

Note, however, that great difficulties aregenerally encountered in a system of the(122) type when we wish to realize practically the Poincare-Bendixson criterion,since no general methods exist for buildingthe appropriate domains and, therefore,success depends both on the type of systemand on the experience of the researcher. Atthe same time we must bear in mind that


finding the conditions in which no limitcycles exist is no less important than establishing criteria for their existence. In thisrespect the most widespread condition is theDulac criterion: if there exists a functionB (.1:, y) that is continuous together uiith. itsfirst partial dericaiives and is such that ina sinlply connected domain D of the phaseplane the sum.

a (BX) + (] (BY)ax Dy

is a function of fixed sign, * then no limitcycles of the differential system (122) canexist in D. At B (x, y) == 1 the criteriontransforms intu the Betulixson criterion.

If we turn to the differential equation(156), which constitutes a differential modelfor describing all adiabatic one-dimeusioualflow of a perfect gas of constant specificheat ratio through a nozzle with drag, thenfor this equation we have

X(x, y)=(1-y)F(x),

y (x, y) =4y (11+ 1';1 y) (Wy-F' (x».

If we put

B (x, y) = {4yF (x) [1 + I' 2 1 y]}-I

* That is, positive definite or negative definite,

16*


we find that

a(BX) + 8 (BY) -.YL > 0iJx iJy F (x)

and, lienee, Eq. (156) has no closed integralcurves.

Let us discuss one more concept that canbe employed to establish the existence oflimit cycles. 'The concept is that of theindex of a singular point.

Let r be a simple closed curve (i.e. acurve without self-intersections) that isnot necessarily a phase trajectory of system(122), Iies in the phase plane, and does notpass through the singular points of thissystem. Then, if 1) (x, y) is a point of I',the vector

V (x, y) == X (x, y) i + Y (x, y) j,

with i and j the unit vectors directed alongthe Cartesian axes, is a nonzero vector and,hence, is characterized by a certain directionspecified by an angle e (Figure 89). Ifpoint P (x, y) moves along I', say, counterclockwise and completes a full cycle, vectorV performs an integral number of cyclesin the process, that is, angle e acquires anincrement L\8 = 2nn, with n a positiveinteger, a negative integer, or zero. Thisnumber n is said to be the index of the closedcurie r (or the index of cycle f).

If we begin to contract r in such a mannerthat under this deformation r does not


y

r

o~--------~~

Fig. 89

pass through singular points of the givenvector field, the index of the cycle must,on the one hand, vary continuously and,on the other, remain an integer. This meansthat under continuous deformation of thecurve the index of the cycle does not change.This property leads to the notion of theindex of a singular point as the index of asimple closed curve surrounding the singular point.

The index has the following properties:(1) the index of a closed trajectory of the

differential system (122) is equal to +1,(2) the index of a closed curve surrounding

several singular points is equal to the sumof the indices of these points, and

(3) the index of a closed curve encompassingonly ordinary points is zero.


This implies, for one thing, that sincethe index of a closed trajectory of system(122) is always +1, a closed trajectory mustenclose either one singular point with anindex +1 or several singular points witha net index equal to +1. This fact is oftenused to prove the absence of limit cycles.

The index of a singular point is calculatedby the formula

e--hn =:: 1 + -2-' (179)

where e is the number of elliptic sectors,and h is the number of hyperbolic sectors.For practical purposes the following simplemethod may be suggested. Suppose that Lis a cycle that does not pass through singular points of (122) and is such that anytrajectory of (122) has no more than a finitenumber of points common to L. The trajectories may intersect L or touch it. Inthe latter case only exterior points of contact (type A) or interior points of contact(type B) are taken into account, while theC-type points, points of inflection, are not(see Figure 90). We can still use formula(179) to calculate the index of a singularpoint, but now e is the number of interiorpoints of contact and h the number ofexterior points of contact of trajectoriesof (122) with cycle L.

Figure 91 depicts singular points with


Fig. 90

247

indices 0, +2, +3, +1, and -2, respectively.

Earlier it was noted that constructinga complete picture of the behavior of thephase trajectories of ths differential system(122) is facilitated by introducing the pointat infinity via transformations (159). Topological considerations provide a verygeneral theorem which states that whena continuous vector field with a finite number of singular points is specified on asphere, the net index of the points is +2.Thus, if the net index of all the singularpoints of a differential system (possessinga finite number of such points) that lie ina finite phase-plane domain is distinct

.from +2, the point at infinity must be asingular point with a nonzero index.

But if instead of inversion we employhomogeneous coordinates, the net index ofall the singular points is already +1. That


this is so can be seen from the fact that ifthe plane is projected onto a sphere withthe center of projection placed at the centerof the sphere, two points on the sphere correspond to a single point on the projectiveplane, and the circumference of the greatcircle parallel to the plane corresponds toa straight line at infinity.

If we turn to Eq. (168), which describesadiabatic one-dimensional flow of a perfectgas through a rotating tube of uniform crosssection, then, as shown in Figure 85, for thesingular point at infinity we have e = 2and h == O. It follows from this that theindex of this point is +2 and does notdepend on the values of the constants inEq. (168). This implies, for one thing, thatthe net index of finite singular points iszero. I t can also be shown that, dependingon whether the straight line specified bythe equation my - Gx2 == 0 has two points,one double point, or not a single point ofintersection with the parabola fixed by theequation 1 - py +qG2x2 = 0 (which, in turn,is equivalent to G being greater than Go'equal to Go, or less than Go' with Go =2mql / 2/p ), the following combinations offinite singular points occur.

(a) G > Go. A saddle point and a nodalpoint.

(b) G = Go. A higher-order singular pointwith two hyperbolic (h = 2) sectors andtwo parabolic (e = OJ.

(c) G < Go. Singular points are absent.We see that in three cases the net index

is zero, as it should be.

2.14 Periodic Modes in ElectricCircuits

We will SIlOW how limit cycles emerge ina dynatron oscillator (Figure 92). An analysis of the operation of SUCIl an oscillatorleads to what is known as the Van der Polequation. Although phenomena linked withgeneration of limit cycles can be illustratedby examples from mechanics, biology, andeconomics,' we will show how SUC]l phenomena emerge in the study of electriccircuits.

Figure 92 depicts schematically a dynatron oscillator, with the ia-va characteristicshown by a solid curve in Figure 93. Hereia is the current and v« the voltage in the

i

c

Fig. 92

Ch. 2.. Qualitative Methods 251

screen-grid tube, the dynatron. The circuitconsists of resistance r, inductance L,and capacitance C connected in paralleland known as the tank circuit, which isconnected in series with the screen-grid tube.The real circuit can be replaced in this casewith an equivalent circuit shown schematically in Figure 94. The characteristic of thetube may be approximated with a thirddegree polynomial i == au + ,\,u3

, which isshown by a dashed curve in Figure 93.Here i and v stand for the coordinates ina system whose origin is shifted to thepoint of inflection O. As follows fromFigure 93,

a >0, '\' >0.In accordance with one of Kirchhoff's laws,

i + i; + iL + i c == 0,with i , == vir, L (diL/dt) == v, and ic =Cv. As a result of simple manipulations wearrive" at the following differential equation:

.. (a 1 3'\' ). 1v + C+rc+C v2 v-f- LC =:;,0.

If we now puta 1 _ 3y _ 1 _ 2c+rc- a,,- c-b~ LC -0)0'

we can write the previous equation in theform

~ ·+ "(~ + bv2) ~ + ro~v = 0.


II

/~

- .....v

Fig, 93

i

cL

+T~---

-1 .L..-...--"'-----------...

Fig. 94

This differential equation is known as theVan der Pol equation. If we introduce thetransformations

v=y,•• dyV= y dV'

we can associate with the Van der Po]

equation the following first-order differential equation:

dy (a+bv2)y+roBv Y(v, y) (180)dU== - y - V (v, y)

The only finite singular point of thisequation is the origin, and

J* (0, 0) = ro~ > 0,

D (v, y) = -(a + bv2) .

Since b > 0, we assume that a > 0 andconclude that divergence D does not reverse its sign and, hence, Eq. (180) canhave no closed integral curves. We, therefore, will consider only the case where ais negative, that is, a < -1/r. This implies that D (0, 0) = -a > 0 and, hence,the singular point is either a nodal pointor a focal point. If we now consider thedifferential system corresponding to thedifferential equation (180), that is,dy dvdt= Y (v, y), dt =. V (v, y),

we see that as t grows, the representativepoint moves along a trajectory toward thesingular point. Thus, a trajectory thatemerges from the point at infinity cannotreach the singular point at the origin nomatter what the value of t including thecase where t = +00. This implies that ifwe can prove that a trajectory originatingat the point at infinity resembles a spiral


winding onto the origin as t ~ 00, thiswill guarantee that at least one Iimit cycleexists.

The existence proof for such a cycle canbe obtained either numerically 01' analytically. A numerical method suggested by theDutch physicist and mathematician Vandel" Pol (1889-1959) consists in buildinga trajectory that originates at a pointpositioned at a great distance Irom theorigin and in checking whether this trajectory possesses the above-mentioned property. Such a procedure yields an approximation to the limit cycle, but it can beemployed only in the case where concretenumerical values are known.

Below we give a proof procedure" basedon analytical considerations and the investigation of the properties of singularpoints at infinity. Here, in contrast to themethod by which Eq, (168) was studied,we eID ploy the Inure convenient (in thepresent problem) hom ugeneous-cuordinatetransformations

v = ~/z, Y == "liz. (18t)

The straight line at infinity is fixed by theequation z = O. To reduce the number of

* See the paper by J. Kestin and S.I(. Zaremba,"Geometrical methods in the analysis of ordinarydifferential equations," Appl. Sci. Res. B3: 144189 (1953).


variables to two, we first exclude the point~ == z == O. This can be done if we assumethat ~ = 1. Then

v == 1lz, Y == '}l/z.

In this case the d ifferential system associated with Eq. (180) is transformed todz _ dn _ (az2+u)11+(O~z2 2(it - - 11Z , dt - - Z2 - 11 •

It is convenient to introduce a new para ..meter, 0, in the following manner:

dt == Z2 dtl. (182)

(183)

Then this system of equa tions can be written as

d11(IO= -(az2 + b) l1-W~z2-f)2z2=ag.,

dz __ 3_~dO- llz -;;,.

Note, first, that the straight line at infinity"z == 0, constitutes a trajectory of the differential system (183) and as t grows (hence,as e grows) the representative point movesalong this trajectory toward the only singular point z = 11 = O. The characteristicequation, which in this case has the form-b1lz = 0, specifies the regular exceptionaldirection z == 0, with two repulsive regionscorresponding to this direction, which canbe established. by studying the appropriate


Fig. 95

vector field (Figure 95). The second exceptional direction 11 == 0 is singular and,therefore, additional reasoning is required.

The locus of points ay = 0 is a curve thattouches the straight line 11 == 0 at theorigin and passes through the second andthird quadrants (Figure 95), which makesit possible to fix three directions, I, II,and I I I, on different sides of the symmetryaxis z == O. The region lying between thecurve fixed by the equation 6!J = 0 andthe axis 11 == 0 is topologically equivalentto two repulsive regions. Thus, on each sideof the straight line II = 0 there is at least

en. 2. Qualitative l\fethods 257

one trajectory that touches the straightline at the origin.

If we now consider the differential equation

dn a b (U~ + lJdi ==z+7+t1Z z=! (11, z),

we note that for small values of I 11 I!J- =-i- ( 1 _ w~ ) < 0Of) Z f)2

in the second quadrant between curve c:!f =o and axis 11 = O. Hence, if we take twophase trajectories with the same value of 11,it is easy to see that the representativepoints moving along these trajectories willalso move apart as z decreases. This meansthat on each side of the straight line 11 = 0there can be only one phase trajectory thattouches this straight line at the originand belongs to the region considered. It isalso clear that the qualitative picture issymmetric about the axis z = O. More thanthat, since for small values of z and positive11 we have

I VjJ/Z ,~ w~/I "lZ I,there can be no curves that touch the axis11 = 0 at the origin and pass through thefirst or fourth quadrant. Such curves arealso absent from the region that lies to theleft of the curve fixed by the equation~ = 0 since in this case vg is positive and

17-0770


% has the same sign as z (Figure 95,arrows IV). This reasoning suggests thatthe singular point 'r) == z == 0 is a saddlepoint. The two phase trajectories that reachthis point as t --+ +00 are segments of thostraight line at infinity, z = 0, that link theprevious point with the point ~ == z == o.The other two phase trajectories reach thesaddle point as t -+ - 00.

In our reasoning we did not discuss thepoint S == z == O. To conclude our investigation, let us put 11 == 1 in (181). Then thedifferential system associated with the differential equation (180) can be written asfollows:

:J =z2+£(az2+bs2+w;sz2)=p,

dzdO::=' 2, (az2 +b~2 + ro~~Z2) = Q,

where the variable 0 is defined in (182). Thepoint ~ == z = 0 proves to be singular, andthe characteristic equation here is Z3 == o.Hence, the exceptional direction z == 0 issingular, too. The curve fixed by the equation P (6, z) == O~ (Figure 96) touches theaxis z = 0 at the origin and has a cuspidalpoint there, while the curve fixed by theequation Q (£, z) == 0 has a multiple pointat the origin. By studying the signs of Pand Q we can establish the pattern of thevector field (Figure 96) and the phase-tra-

;.Ch. 2. Qualitative Methods

tl=O\

P=O --r\ \,,~ \

,1"'-_\

/r"""'-- // /

/I.r

Fig. 96

258

Fig. 97

jectory pattern in the neighborhood of thesingular point ~ = z = 0 (Figure 97).

We note, for one thing, that far from theaxis ~.== 0 there are no phase trajectoriesthat enter the singular point from the right.17*


This follows from the fact tha t in the firstand fourth quadrants

I z/~ I > IQIP I·The above reasoning shows that the Van

der Pol equation has no phase trajectoriesthat tend to infinity as t grows but has aninfinitude of phase curves that leave infinityas t grows. This proves the existence of atleast one limit cycle for tile Van der Poldifferential equation.

2.15 Curves Without Contact

In fairly simple cases the complete patternof the integral curves of a given differentialequation or, which is the same, the phasetrajectory pattern of the correspondingdifferential system is determined by thetype of singular points and closed integralcurves (phase trajectories), if the latterexist. Sometimes the qualitative picturecan be constructed if, in addition to establishing the types of singular points, we canfind the curves, separatrices, that link thesingular points. Unfortunately, no generalmethods exist for doing this. Hence, it isexpedient in qualitative integration toemploy the so-called curves without contact. The reader will recall that a curve oran arc of a curve with a continuous tangentis said to be a curve (arc) without contactif it touches the vector (X, Y) specified

Ch. 2. Qualitat ive Methods 261

by the differential system (122) nowhere.The definition implies that vector (X, Y)must point in one direction on the entirecurve. Thus, a curve without contact maybe intersected by the phase trajectories of(122) only in one direction when t growsand in the opposite direction when t diminishes. Therefore, knowing the respectivecurve may provide information about thepattern of a particular part of a phase trajectory.

Various auxiliary inequalities can beused in qualitative integration of differential equations. For example, if two differential equations are known, say,

dy )dx == f (z, y ,

dlld~ =:: g (x, y)

and it is known that f (x, y) ~ g (x, y)in a domain D, then, denoting by Yt (x)a solution of the first equation such thatYl (xo) = Yo, with (xo, Yo) ED, and byY2 (x) a solution of the second equationwith the same initial data, we can provethat YI (x) ~ Y2 (x) for x ~ X o in D. Butif in D we have the strict inequalityf (x, y) < g (x, y), then Yl (x) < Y2 (x) forx > Xo in D and the curve y == Y2 (x) isa curve without contact.

As an example let us consider the differential equation (168). We have already

shown that if G > Go, the equation allowsfor two finite singular points, a saddlepoint and a nodal point, which are pointsof intersection of the straight line myG2X == 0 and the parabola 1 - py +qG2x 2 == O. The segments of the straight lineand parabola that link these two finite singular points are curves without contact andthey specify a region of the plane which wedenote hy ~. If we put

x (x, y) = 1 - py + qG2x2 ,

y (x, y) = 2y (lny - G2.X),

it is easy to see that vee tor (X, Y) is elirected on the boundary of 11 outwardexcept at the singular points. Hence, if therepresentative point emerges from any interior point of 11 and moves along an integralcurve with t decreasing, it cannot leave Awithout passing through one of the singularpoints. But since inside 11 we haveX (x, y) < 0, the singular point that isattractive is the nodal point.

Finding the slopes of the exceptionaldirections for the saddle point, we can seethat one of the exceptional straight linespasses through 11. This implies that anintegral curve that is tangent to thisstraight line at the saddle point must enter~ and then proceed to the nodal point.

ell. 2. Qualitative Methods 2G3

2.16 The Topographical Systemof Curves. The Contact Curve

Earlier we noted that in the qualitativesolution of differential equations it is expedient to use curves without contact. I t mustbe noted, however, that there is no generalmethod of building such curves that wouldhe applicable in every case. Hence, theimportance of various particular methodsand approaches in solving this problem.One approach is linked to the selection ofan appropriate topographical system ufcurves. Hero a topographical systeln of curiesdefined by an equation Q> (x, y) == C, withC a real-valued parameter, is understood tobe a falnily of nonintersecting; envelopingeach other, and continuously differentiablesimple closed curves that completely fill adoubly connected domain G in the phaseplane.*

If the topographical system is selectedin such a manner that each value of parameter C is associated with a unique curve,then, assuming for the sake of definitenessthat the curve corresponding to a definitevalue of C envelopes all the curves withsmaller values of C (which means that the"size" of the curves grows with C) and thatno singular points of the corresponding

• Other definitions of a topographical system alsoexist in the mathematical literature, but essentiallythey differ little from the one given here.

differential system lie on the curves belonging to the topographical system, wearrive at the following conclusion. If weconsider the function <D (<p (t), 'I' (t» wherex = (p (t) and y = '¢ (t) are the parametricequations of the trajectories of the differential system of the (122) type and calculatethe derivative with respect to t, that is,

d<l) (/dtis a function of fixed sign. If, in addition,d<D/dt is positive on a certain curve belonging to the topographical system, thensuch a curve, being a cycle without contact,possesses the property that all phase trajectories intersecting it leave, with thepassage of time t, the finite region boundedby it. And if d<D/dt is negative, all thephase trajectories enter this region. Thisimplies, for one thing, that if in an annularregion completely filled with curves belonging to the topographical system the deriva tive d(D/dt is a function of fixed sign,the region cannot contain any closed trajectories, say, the limit cycles of the differential system. Limit cycles may existonly in the annular regions where the

derivative d<D/dt is a function with alternating signs.

To illustrate this reasoning, we considerthe differential system

dz 3 dy 3 (184dt==Y_ox+x, dt==-x-y+y, )

which has only one singular point in thefinite part of the plane, 0 (0, 0), a stablefocal point. For the topographical systemof curves we select the family of concentriccircles centered a t point 0 (0, 0), that is,the family of curves specified by the equa-tion x2 + y2 == C, with C a positive parameter. In our ease the derivative d/dtis given by the relationshipd<l>d"t:=' - 2 (x2+ y2) + 2 (x~ -t- y4).,.

which in polar coordinates x = r cos 8 andy = r sin e assumes the form

~~ =. - 2r2 + 2r!" (cos!" a+sin!" 0)= - 2r2

+2r4( ~ +i- cos 49).

Noting that the greatest and smallest valuesof the expression in parentheses are 1 and1/2, respectively, we arrive a the followingconclusion. For r greater than Vi the valueof the derivative d<D/dt is positive whilefor r less than unity it is negative. On thebasis of the Poincare-Bendixson criterion

18-0770


we conclude (if we replace t with -t insystem (184)) that the annular region bounded by the circles x2 + y2 = 1 and x2 +y2 = 2 contains a limit cycle of system(184), and this limit cycle is unique.

To prove the uniqueness it is sufficientto employ the Dulac criterion for a doublyconnected domain: if there exists a functionB (x, y) that is continuous together withits first partial derivatives and is such thatin a doubly connected domain G belongingto the domain of definition of system (122)the function

a(BX) + fJ (BY)ax iJy

is of fixed sign, then in G there can be nomore than one simple closed curve consistingof trajectories of (122) and containing withinit the interior boundary of G.

In our case, selecting B (x, y) == 1 forsystem (184), we find that ax/ox +8Y loy = 3 (x2 + y2) - 2 and, as can easilybe seen, in the annular region with boundaries x2 + y2 = 1 and x2 + y2 = 2 the'expression 3 (x2 + y2) - 2 always retainsits sign. Allowing now for the type of thesingular point 0 (0, 0), we conclude thatthe cycle is an unstable limit cycle.

To return to the problem of buildingcurves without contact in the general case,let us examine a somewhat different way

Ch, 2. Qualitative Methods

of using the topographical system ofcurves. This approach is based on thenotion of a contact curve. Introduction ofthis concept can be explained by the factthat if the derivative d(})/dt vanishes ona set of points of the phase plane, this setconstitutes the locus of points at which thetrajectories of the differential SYSlCll1 touchthe curves belonging to the topographicalsystem. Indeed, the slope of the tangent toa trajectory of the differential system isY/X and the slope of the tangent to a curvehelongir-g to the topographical system is-(oCT ox)/(o<t>/ay). 1'hus, when

atl x+ 8$ Y== 0 (185)iJx ay ,

the slopes coincide, or

~ _ _ a<t> I (J(»

X - ax. fJy •

Hence, the locus of points at which thetrajectories of the differential system of the(122) type touch the curves belonging to thetopographical system (defined by the equation <l> (.1, y) = C) is called a contact curve.The equa lion of a contact curve has the(185) form. Of particular interest here isthe case where the topographical systemcan be selected in such a manner that eitherthe contact curve itself or a real branch ofthis curve proves to be a simple closedcurve. Then the topographical system con-\,)*

268 Ditterential Equations in Applications

F"ig. 98

tains the "greatest" and "smallest" curvesthat are touched either by a contact curveor by a real branch of this curve (in Figure 98 the contact curve is depicted by adashed curve). If the derivative dcD/dtis nonpositive (nonnegative) on the "greatest" curve specified by the equation<D (x, y) = C1 and nonnegative (nonpositive) on the "smallest" curve specified bythe equation <D (x, y) = C2 , then the annular region bounded by these curves containsat least one limit cycle of the differentialsystem studied.

Specifically, in the last example (see(184)), the contact curve specified by theequation x2 + y2 = x4 + u' is a closedcurve (Figure 99), which enables us to findthe "greatest" and "smallest" curves in theselected topographical system that aretouched by the contact curve. Indeed, if

Ch. 2. Qualitative Me~ods

Fig. 99

269

we note that in polar coordinates the equation of the contact curve has the form r2 =(cos! e + sin" 8)-1, we can easily find theparametric representation of this curve:

cos a sin ax = , y -= ---;~=:;=:::;::::::::;::::;;-Vcos! e+sin4 e - Vcos- e+sin4 e

Allowing now for the fact that the greatestand smallest values of r 2 are, respectively,2 and 1, we conclude that the "greatest"and "smallest" curves of the topographical


system x2 + y2 = C are the circles x2 +y2 = 1 and x2 + y2 = 2, respectively.These circles, as we already know, specify anannular region containing the limit cycleof system (184).

2.17 The Divergence of a VectorField and Limit Cycles

Returning once more to the general case,we note that in building the topographicalsystem of curves we can sometimes employthe right-hand side of system (122). Ithas been found that the differential systemmay be such that the equation

ax + BY ="Aax ay , (186)

where A is a real parameter, specifies thetopographical system of curves.

For instance, if we turn to the differentialsystem (184), Eq. (186) assumes the form3 (x2 + y2) - 2 = A. Then, assuming thatA = (A + 2)/3, with 'A E (-2, +00), wearrive at the topographical system of curvesx 2 + y2 = A used above.

A remark is in order. If the "greatest" and"smallest" curves merge, that is, the contactcurve or a real branch of this curve coincides with one of the curves belonging tothe topographical system, such a curve isa trajectory of the differential system,


Let us consider, for example, the differential system (173):dxdt ==- - y + x (1 - x 2 - y2),

dydt= x-I·· y (1- x2 - y2).

For this systemax ayax +ay==- 2- 4 (x2+ y2),

and Eq. (186) assumes the form 24 (x2 + y2) == A. If instead of parameter ').."we introduce a new parameter A =(2 - ')..,,)/4, with A E (-00, 2), then thelatter equation, which assumes the formx 2 + y2 == l\, defines the topographicalsystem of curves. The contact curve in thiscase is given by the equation (x2 + y2) X(x2 + y2 - 1) = 0 and, as we see, its realbranch x 2 + y2 = 1 coincides with one ofthe curves belonging to the topographicalsystem. This branch, as shown on pp. 237239, proves to be a limit cycle of system(173).

The last two examples, of course, illustrate, a particular situation. At the sametime the very idea of building a topographical system of curves by employing theconcept of divergence proves to be fruitfuland leads to results of a general nature.We will not dwell any further on these results, but only note that the followingfact may serve as justification for what


has been said: it a differential system of the(122) type has a limit cycle L, there exists areal constant A, and a positive and continuously differentiable in the phase plane function B (x, y) such that the equation

a(BX) + a(BY) A{}z ay

specifies a curve that has a finite real branchcoinciding with cycle L.

Let us consider, for example, the differential system

:: = 2x - 2x3 + x2.y - 3xy2.+ y3,

~:=. _xS + X2y + xy2de .,

which has a limit cycle specified by theequation x2 + y2 = 1. For this system thedivergence of the vector field isax ayiJx +ay=- 2- 5x2

- 3y2.

\\'e can easily verify that there is not asingle real A for which the equation 2 5x2 - 3y2 = A specifies a trajectory of theinitial differential system. But if we takea function B (x, y) = 3x2 - 4xy + 7y2 +3, then

(}(BX) + fJ(BY) (x2 ' 2-1)ax 8y +Y

X (- 23;r2+ 16xy - 25112 - 20)-14,

Ch, 2. Qualitative Methods 273

and the equation

a(BX) -1_ rJ (BY) == _ 14ax I rJy

specifies a curve whose finite real branchx2 + y2 - 1 = 0 proves to be the limitcycle.

In conclusion we note that in studyingconcrete differential models it often provesexpedient to employ methods not discussedin this book. Everything depends on thecomplexity of the differential model, onhow deep the appropriate mathematicaltools are developed, and, of course, on theerudition and experience of the researcher.

Selected Readings

Amel'kin, V.V., and A.P. Sadovskii, MuthematicalModels and D ifJerential Equations (M insk:Vysheishaya Shkola, 1982) (in Russian).

Andronov, A.A., A.A. Vitt, S.E. Khalkin, andN.A. Zheleztsov, Theory of Oscillations (Reading, Mass.: Addison-Wesley, 1966).

Braun, M., C.S. Cleman, and D.A. Drew (eds.),D ifferential E quation Models (New York:Springer, 1983).

Derrick, W.R., and S.I. Grossman, ElementaryDifferential Equations with Applications, 2nded. (Reading, Mass.: Addison-Wesley, 1981).

Erugin, N°.P., A Reader in the General Course ofDifferential Equations (Minsk: Nauka i Tekhnika, 1979) (in Russian).

Ponornarev, K. K. , Constructing D iffereniial E quations (Minsk: Vysheishaya Shkola, 1973) (inRussian).

Simmons, G.F., Differential Equations with Applications and Historical Notes (New York:McGraw-Hill, 1972).

Spiegel, M.R., Applied Differential Equations(Englewood Cliffs, N.1.: PreDtice-I-Iall, 1981).

Appendices

Appendix i,Derivatives of ElementaryFunctions

Function Derivative

C (constant) 0x 1xn IlXn - 1

1 1x -X21 It

xn - x1t +1

Il x t2]1;-

Vx 1

nVxn-1

eX eX

aX aX Ina

In x1x

loga x1 1

- loga e=--x x In a

IOglO x1 0.4343-logloe~--x x

sin x cos xcos x -sinx

276

Function

tan x

cot x

secx

esc x

sin-1 x

cos-1 x

tan-1 x

c.OL-l x

sce-1 :t

CSC-1 x

sinh x

cosh z

tanh x

coth x

sinh-1 x

Appendices

Derivative

1---=sec2 xcos2 x

1--.--= -csc2xSln 2 x

sin zcos2 x = tan x sec x

_ ~os x = _ eot x esc xSln2 x

1

1/1-x2

11-1-- x 2

1-. - 1-l-x 2

1

x "Jlx 2 - 1cosh :vsinh x

1cosh2x

1- sinh2x

1

.i .. _ ..· ...._~..... ~.£Jo.a ...._ ...._ •

Appendices

Function

COSh-1 .1;

tanh-1 x

coth? x

Derivative

1----, x>t]/x2 - 1

11-x2 ' Ix 1< 1

11 _ x2 ' 1·1: I> 1

277

Appendix 2.Basic Integrals

Power ...Law Functions\ xn+1

J xndx= n+ 1 (n =1= 1)

~ :x=ln 1x l

Trigonometric Functions

~ sinxdx=-cosx

~ cos x dx = sin x

~ tan x dx = -In I cos x I

~ cotxdx=ln I sinx I

r ~=f.anxJ cos x

\ ~=-cotxJ Sln2 x

Fractional Rational Functions

Appendices

\ ~==-~cOth-l (~)J x2-a2 a a1 x-a

=--=--In-- (I z ] >a)2a x+a

Exponential Functions

~ eX dx= eX

~ aX dx= I::Hyperbolic Functions

~ sinh x dx= cosh x

~ cosh x dx= sinh x

~ tanhxdx=In I cosh x I

~ coth x dx = In I sinh x 1

~dx-ho =tanhxcos l5IX

~dx

-.--= -cothxsinh'' x

Irrational Functions

279

~ dz · h 1 ( X )---=810 - -ya2-x2 a

~dx ( x ) ,i-,i slnh"! - =lnl (x+ Y x'+a2)1

ya2+x2 a

\' dx = coslr? (-=- ) = In I (x+ -Vxt - a2) IJ yX2-a2 a

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Differential Equations in Applications Amelkin Wc

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