LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University.

LURE 2009 SUMMER PROGRAMLURE 2009 SUMMER PROGRAMJohn AlfordJohn Alford

Sam Houston State UniversitySam Houston State University

Some Theoretical ConsiderationsSome Theoretical Considerations

Differential Equation ModelsDifferential Equation Models

A first-order A first-order ordinary differential ordinary differential equationequation (ODE) has the general (ODE) has the general formform

),( xtfdt

dx


A first-order ODE together with an A first-order ODE together with an initial conditioninitial condition is is called an initial called an initial value problem (IVP)value problem (IVP)..

ODEODE INITIAL CONDITIONINITIAL CONDITION

00 )( ),(/ xtxxtfdtdx


When there is no explicit When there is no explicit dependence on dependence on tt, the equation is , the equation is autonomousautonomous

Unless otherwise stated, we now Unless otherwise stated, we now assume autonomous ODEassume autonomous ODE

)(xfdt

dx


We may be able to We may be able to solvesolve an an autonomous ode by autonomous ode by separating separating variables (see chapter 9.1 and 9.2 variables (see chapter 9.1 and 9.2 in Thomas’ calculus textbook!)in Thomas’ calculus textbook!)

– separateseparate

dtdxxf

xfdt

dx1

)(

1)(


– integrateintegrate

dtdxxf

1)(

1


A A linearlinear autonomous IVP has the autonomous IVP has the formform

(*)(*)

where where aa and and bb are constants are constants

0)0( , xxbaxdt

dx


The solution of (*) isThe solution of (*) is

(You should check this)(You should check this)

Is this the only solution?Is this the only solution?

a

bebaxtx

at

)()( 0

Differential Equation ModelsDifferential Equation ModelsExistence and Uniqueness Theorem for an IVPExistence and Uniqueness Theorem for an IVP

Differential Equation ModelsDifferential Equation Models Example of Example of non-uniquenessnon-uniqueness of solutions of solutions

It is easy to check that this IVP has a It is easy to check that this IVP has a constantconstant solution solution

0)0( ,3/1 xxdt

dx

ttx allfor 0)(

Differential Equation ModelsDifferential Equation Models Others? (separate variables)Others? (separate variables)

After integrating both sidesAfter integrating both sides

dtdxxxdt

dx 1 3/13/1

2

3 3/2 Ctx

Differential Equation ModelsDifferential Equation Models Must satisfy initial conditionMust satisfy initial condition

Solve for Solve for xx to get to get anotheranother solution solution to the initial value problemto the initial value problem

2

3 0 0)0( 3/2 txCx

3

2)(

3/2

ttx


Which path do we choose if we start from Which path do we choose if we start from t=0t=0??


Existence and uniqueness theorem Existence and uniqueness theorem does not tell us does not tell us howhow to find a to find a solution (just that there is one and solution (just that there is one and only one solution)only one solution)

We could spend all summer talking We could spend all summer talking about how to solve ODE IVPs (but about how to solve ODE IVPs (but we won’t)we won’t)


We might sayWe might say– A fixed point is A fixed point is locally stablelocally stable if starting close if starting close

(enough) guarantees that you stay close.(enough) guarantees that you stay close.

– A fixed point is A fixed point is locally asymptotically stablelocally asymptotically stable if if all sufficiently small perturbations produce all sufficiently small perturbations produce small excursions that eventually return to the small excursions that eventually return to the equilibrium.equilibrium.


In order to determine if an equilibrium In order to determine if an equilibrium x*x* is locally asymptotically stable, letis locally asymptotically stable, let

to getto get

the the perturbation equationperturbation equation

*)()( xtxt

)( )( xfdt

dxf

dt

dx


Use Use Taylor’s formulaTaylor’s formula (Cal II) to expand (Cal II) to expand f(x)f(x) about the equilibrium (assume about the equilibrium (assume f f has has at least two continuous derivatives with at least two continuous derivatives with respect to respect to xx in an interval containing in an interval containing xx*)*)

where is a number between where is a number between xx and and xx* * and prime on and prime on ff indicates derivative with indicates derivative with respect to respect to xx

why?why?

2/ )('' )(' )()(2 **** xxfxxxfxfxf


Use the following observationsUse the following observations

andand

to getto get

why?why?

0)( * xf

0 small 2* *xxxx

)(' )(' )( *** xfxxxfxf


Thus, assuming smallThus, assuming small

yields that an approximation to the yields that an approximation to the perturbation equationperturbation equation

is the equationis the equation

why?why?

*xx

)(' * xf

dt

d

)( xf

dt

d


The approximationThe approximation

is called the is called the linearizationlinearization of the original of the original ODE about the equilibriumODE about the equilibrium

why?why?

)(' * xf

dt

d


Let and assumeLet and assume

TwoTwo types of solutions to linearization types of solutions to linearization

– decayingdecaying exponential exponential

– growinggrowing exponential exponential

why?why?

tedt

d 0

)(' *xf 0

0

0


Fixed Point Stability TheoremFixed Point Stability Theorem


Application of stability theorem:Application of stability theorem:

Fixed points:Fixed points:

0 ,0 ,1

Kr

K

xrx

dt

dx

KxxK

xrxxf

,0 01)(


Differentiate Differentiate ff with respect to with respect to xx

Substitute fixed pointsSubstitute fixed points

K

xrxf

21)('

0)(' and 0)0(' rKfrf


Fixed Point Stability Theorem showsFixed Point Stability Theorem shows

– x=0x=0 is unstable and is unstable and x=Kx=K is stable is stable

NOTICE: stability depends on the NOTICE: stability depends on the parameter parameter rr!!


A A GeometricalGeometrical (Graphical) Approach to (Graphical) Approach to Stability of Fixed PointsStability of Fixed Points

– Consider an Consider an autonomousautonomous first order ODE first order ODE

– The The zeroszeros of the graph for of the graph for

are the fixed points are the fixed points

)(xfdt

dx

xxf vs.)(


Example:Example:

Fixed points:Fixed points:

)1( xxdt

dx

1 and 0 *2

*1 xx


Graph f(x) vs. x


)(xfdt

dx


Imagine a particle which moves along Imagine a particle which moves along the the xx-axis (one-dimension) according to-axis (one-dimension) according to

particle moves right particle moves right

particle moves leftparticle moves left

particle is fixedparticle is fixed

This movement can be shown using arrows This movement can be shown using arrows on the x-axison the x-axis

)(/ xfdtdx

0)( xf 0)( xf 0)( xf


Last graphLast graph

right) (arrows 0)(

and *4

*3

*2

*1

xf

xxxxxx

left) (arrows 0)(

and , , *4

*3

*2

*1

xf

xxxxxxx


Theorem for local asymptotic stability of Theorem for local asymptotic stability of a fixed point used the sign of the a fixed point used the sign of the derivative of derivative of f(x)f(x) evaluated at a fixed evaluated at a fixed point:point:

at stability asymptotic local 0)(' ** xxf

at y instabilit 0)(' ** xxf


Last graphLast graph

– are are unstableunstable because because

– are are stablestable because because

0)(' and 0)(' *3

*1 xfxf

0)(' and 0)(' *4

*2 xfxf

, *3

*1 xx

, *4

*2 xx


Fixed points that are locally Fixed points that are locally asymptotically stable are denoted with a asymptotically stable are denoted with a solid dot on the solid dot on the xx-axis-axis

Fixed points that are unstable are Fixed points that are unstable are denoted with an open dot on the denoted with an open dot on the xx-axis.-axis.


Putting the arrows on the Putting the arrows on the xx-axis along -axis along with the open circles or closed dots at with the open circles or closed dots at the fixed points is called plotting thethe fixed points is called plotting the phase linephase line on theon the x-axis x-axis

Bifurcation TheoryBifurcation Theory

How Parameters Influence Fixed PointsHow Parameters Influence Fixed Points


Example equationExample equation

Here Here aa is a real valued parameter is a real valued parameter Fixed points obeyFixed points obey

2xadt

dx

axxa 22 0


0a


Fixed points depend on parameter Fixed points depend on parameter aa

i) two stablei) two stable

ii) one unstableii) one unstable

iii) no fixed points existiii) no fixed points exist 0a

0 0 * xa

and 0 *2

*1 axaxa


The parameter values at which The parameter values at which qualitative changes in the dynamics qualitative changes in the dynamics occur are called occur are called bifurcation pointsbifurcation points..

Some Some possiblepossible qualitative changes qualitative changes in dynamicsin dynamics

– The number of fixed points changeThe number of fixed points change– The stability of fixed points changeThe stability of fixed points change


In the previous example, there was In the previous example, there was a bifurcation point at a bifurcation point at a=0a=0..

– For a>0 there were two fixed pointsFor a>0 there were two fixed points– For a<0 there were no fixed pointsFor a<0 there were no fixed points

When the number of fixed points When the number of fixed points changes at a parameter value, we changes at a parameter value, we say that a say that a saddle-node bifurcationsaddle-node bifurcation has occurred.has occurred.


Bifurcation DiagramBifurcation Diagram– fixed points on the vertical axis and fixed points on the vertical axis and

parameter on the horizontal axisparameter on the horizontal axis– sections of the graph that depict sections of the graph that depict

unstable fixed points are plotted unstable fixed points are plotted dasheddashed; sections of the graph that ; sections of the graph that depict stable fixed points are depict stable fixed points are solidsolid

– the following slide shows a bifurcation the following slide shows a bifurcation diagram for the previous examplediagram for the previous example


Example equationExample equation

Here Here aa is a real valued parameter is a real valued parameter Fixed points obeyFixed points obey

)1ln( xxadt

dx

? 0)1ln( xxxa


DefineDefine

ThenThen

xxxfaxf )1ln()( and )( 12

)()( 12 xfxfdt

dx


Fixed points obeyFixed points obey

For different values of For different values of aa, graph each , graph each function on the same grid and function on the same grid and determine if graphs intersect. The determine if graphs intersect. The xx-values at intersection (if any) are -values at intersection (if any) are fixed points.fixed points.

)()( 0)()( 1212 xfxfxfxf


aa= 1= 1


aa= 0= 0


aa= -1= -1


From graphical analysis, there From graphical analysis, there appear to be three appear to be three qualitativelyqualitatively different casesdifferent cases

– a>0 no fixed pointsa>0 no fixed points– a=0 one fixed pointa=0 one fixed point– a<0 two fixed pointsa<0 two fixed points

A A saddle-nodesaddle-node bifurcation occurs at bifurcation occurs at the bifurcation value the bifurcation value aa=0=0


Stability can be determined Stability can be determined graphically also by plotting the graphically also by plotting the phase line (direction arrows along phase line (direction arrows along the the xx-axis) using the sign of the -axis) using the sign of the right side of the oderight side of the ode

)()( 12 xfxfdt

dx


Arrows point Arrows point rightright when graph 2 is when graph 2 is aboveabove graph 1 graph 1

Arrows point Arrows point left left when graph 2 is when graph 2 is belowbelow graph 1 graph 1

)()( 12 xfxf

)()( 12 xfxf


Stability can also be determined Stability can also be determined using (local asymptotic) stability using (local asymptotic) stability theorem (do the calculus!)theorem (do the calculus!)


First, differentiateFirst, differentiate

After a little algebraAfter a little algebra

x

xxadx

dxf

1

11)1ln()('

x

xxf

1)('


If If aa<0, there are <0, there are twotwo fixed points fixed points– The one on the left is stable sinceThe one on the left is stable since

– The one on the right is unstable sinceThe one on the right is unstable since

0)(' 01 *1

*1 xfx

0)(' 0 *2

*2 xfx


OK- LURE students, what does the OK- LURE students, what does the bifurcation diagram look like??bifurcation diagram look like??

(see next slide)(see next slide)


Use Strogatz’s Nonlinear Dynamics and Chaos Use Strogatz’s Nonlinear Dynamics and Chaos to learn about the following bifurcationsto learn about the following bifurcations

– A) transcritical bifurcation (pg. 50-52)A) transcritical bifurcation (pg. 50-52)

(do problem 3.2.4 on page 80)(do problem 3.2.4 on page 80)– B) pitchfork bifurcation (pg. 55-60)B) pitchfork bifurcation (pg. 55-60)

(do problem 3.4.3 on page 82)(do problem 3.4.3 on page 82)– C) imperfection bifurcations, C) imperfection bifurcations,

catastrophes (pg. 69-73) catastrophes (pg. 69-73)

(do problem 3.6.2 (a) and (b) only on (do problem 3.6.2 (a) and (b) only on page 86)page 86)

LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University.

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