Physics 4550, Fall 2003 – Dynamical Systems 6

Notes on Dynamical Systems (continued)

2. Maps

The surprisingly complicated behavior of the physical pendulum, and many other physicalsystems as well, can be more readily understood by examining their discrete time versions. The fact isthat observations of change are always recorded by sampling systems at discrete moments. Thus, whilewe imagine that we can approximate the actual behavior of a system evolving continuously in time bysampling very rapidly, we often are stuck with a fastest sampling rate fixed by the clock speed or memorylimitations of the computer doing the sampling. So let’s reexamine some of the examples givenpreviously in terms of discrete time. In all of the following we assume that the instants at which thesystem we are interested in are sampled are given by

†

tn = nDt , where

†

Dt is the sampling interval—thetime between samples. (We assume that we sample at a constant rate.)

Example #5: Exponential dynamics: First let’s return to the example of a constant force applied to aviscously damped mass. The velocity of this mass at any instant was previously stated to be

†

v(t) = vT + (v0 - vT )exp(-bt / m) . Let

†

v(tn ) be denoted by

†

vn. The quantity

†

m b has the dimensions oftime and when t increases by

†

m b, v decreases by a factor of 1/e. Thus,

†

m b is called the e-folding time.Let’s assume that our sampling rate is fixed so that N samples are acquired every

†

m b. In other words, weassume that

†

Dt = (m b) N . If we insert this in our expression for v we get

†

vn = vT + (v0 - vT )exp(-n / N ) .

Now, let’s find how the next v is related to the present one:

†

vn+1 = vT + (v0 - vT )exp(-[n +1] / N )

= vT + (v0 - vT )exp(-n / N )exp(-1 N )

= vT + (vn - vT )exp(-1 N )

= avn + b

where

†

a = exp(-1 N ) and

†

b = vT (1- exp[-1 N ]) . In the language of discrete dynamical systems, anequation relating future values of the dynamical state at discrete times to past values is called a map. Themap

†

vn+1 = avn + b is said to be one-dimensional because the next value of v depends only on the currentvalue and no others.

Example #6: Periodic dynamics: Consider the periodic state

†

v(t) = vmax cos(wt +f). Let

†

Dt = 2p wN .Then

†

vn = vmax cos(2pn N +f)

and

†

vn±1 = vmax cos(2p[n ±1] / N +f )

= vmax[cos(2pn N +f)cos(2p N ) m sin(2pn N +f )sin(2p N )]

Adding

†

vn+1 to

†

vn-1 and rearranging terms lead to

Physics 4550, Fall 2003 – Dynamical Systems 7

†

vn+1 = avn - vn-1

where

†

a = 2cos(2p N ) . The map

†

vn+1 = avn - vn-1 is two-dimensional because the next value of vdepends on two earlier ones (not one, as in the previous example).

3. Graphical iteration and fixed points

Irrespective of how successive values of a dynamical process are related, a plot of

†

vn as afunction of n is called a time series plot and a plot of

†

vn+1 as a function of

†

vn is called a first return plot.The first return plot of one-dimensional dynamics is sufficiently simple that graphical iteration can beused to get a sense of what the dynamics is capable of producing. Let’s look at exponential dynamics as aspecific example. Make a graph with the

†

vn+1 along the y-axis and

†

vn along the x. Draw on the graph the dynamical rule

†

vn+1 = avn + bas well as the 45˚ line

†

vn+1 = vn . Graphical iteration is a way ofdetermining successive values of v by drawing. Pick a starting value(v0, v0). Draw a vertical line from that point to the curve (line)representing the dynamical rule. The y-value of that point is v1.Draw a horizontal line from the latter point until it hits the 45˚ line.The new point on the 45˚ line has coordinates (v1, v1). Repeat(iterate). The figure to the right shows an example, where

†

a ispositive and less than 1, and

†

b is positive. (In the language ofdynamical systems

†

a and

†

b are called control parameters.)

The point of intersection between the dynamics and 45˚ lines is special. If the starting value wereexactly there, there would be on place to iterate to. Such a constant value is called a fixed point. Forexponential dynamics, the fixed point, vf, is easy to calculate:

†

vf = avf + b fi vf =b

1-a=

vT (1- exp(-1 N )1- exp(-1 N )

= vT .

The fixed point is the terminal velocity. That’s reassuring. You can see from the figure above that thestarting velocity (“0”) is less than vf and as time goes on it gradually increases to it. Try graphicallyiterating for a starting value greater than vf. Do you get what you expect?

It is instructive to allow

†

a and

†

b in the dynamicsrule to have any values. Contrast the situation shown in theprevious figure with the case to the right where

†

a > 1 and

†

b < 0. Obviously, the starting value now “runs away” fromthe fixed point. Thus, in the previous case, the fixed point isan attractor while in the case shown to the right the fixedpoint is a repellor. These cases are also sometimes said torepresent stable and unstable equilibria, respectively. (Thebottom of a bowl is a stable equilibrium point for a marble ifthe bowl is open side up and the marble is inside. Smalldisplacements of the marble to either side of the bottom leadto returns. On the other hand, if the open side is down and themarble is perched on top that spot is an unstable equilibrium.Small displacements now run away from the top.) The technical difference between stable and unstable

v(n+1)

v(n)

0

1 2

dynamics line

45˚ line

v(n+1)

v(n)

0

1

2

dynamics line 45˚ line

Physics 4550, Fall 2003 – Dynamical Systems 8

fixed points is the slope of the dynamics rule at the fixed point. If the slope has a magnitude less than 1,the fixed point is stable. If the slope has magnitude greater than 1, the fixed point is unstable. (Whathappens if the slope has a magnitude exactly equal to 1?)

4. One-dimensional, nonlinear dynamics

The most famous one-dimensional, nonlinear dynamical system is the logistic map:

†

vn+1 = a(1- vn )vn , (5)

where usually one uses values

†

0 £ a £ 4 and

†

0 £ vn £ 1. If the stuff inside the parentheses in (5) wereequal to 1, then (5) would just describe simple exponential dynamics (with

†

b = 0). The addition of the

factor

†

(1- vn ) makes an amazing difference. A first return plot derived from (5) is a parabola that is zeroat both vn = 0 and = 1, and that has a maximum at vn = 1/2. The maximum height of this parabola is

†

a 4.(Thus, to keep

†

vn between 0 and 1,

†

a has to be between 0 and 4. Should

†

vn ever be greater than 1, then

†

vn+1 would be < 0, and so would all successive values. Indeed,

†

vn Æ -•, once this occurs.) The fixedpoints of (5) are

†

vf = a(1- vf )v f fi vf = 0 and vf =a -1

a

The slopes of the tangent lines to the parabola at the fixed points determine their stability. The slope at thefixed point 0 is

†

a , and at the nonzero fixed point it is

†

(2 -a ) a . When

†

a is less than 1, the slope at 0 ispositive and less than 1, so the fixed point at 0 is stable. On the other hand, the nonzero fixed point isunstable because the slope there is greater than 1. The situation reverses as

†

a increases beyond 1. For

†

aslightly larger than 1, the slope at 0 is greater than 1, but the slope at the nonzero fixed point is less than 1.The dynamics is said to bifurcate at

†

a = 1, meaning that it makes a transition from having an attractor at0 to having one at some positive value. Once

†

a is greater than 2, the slope at the nonzero fixed point goesnegative. Its magnitude is still less than 1 until

†

a = 3. For values of

†

a between 3 and 4, however, bothfixed points are unstable. So what happens to the dynamics then?

To see what occurs it is useful to consider higher order return plots. For example,

†

vn+2 = a(1- vn+1)vn+1 = a[1-a(1- vn )vn][a(1- vn )vn]

This expression is quartic in vn. Because of this it has four fixed points. These fixed points correspond tostates whose values repeat every two samples. They are roots of the polynomial equation

†

vf = a[1-a(1- vf )v f ][a(1- v f )vf ] . As the fixed points of the first order map repeat every time, they

obviously also repeat every two times. So two of the four fixed points are 0 and

†

(a -1) a . The other twocan be obtained by dividing

†

vf -a[1-a(1- vf )v f ][a(1- v f )vf ] = 0 by

†

vf [vf - (a -1) a] . The result is

†

vf ± =12

+1

2a±

12a

(a - 3)(a +1) . (6)

If we plug

†

vf + into the first order logistic map, (5), we get

†

vf - out, and vice versa. In other words, the

two roots (6) form a two-cycle—two values that alternate back-and-forth between each other. You can seeimmediately that if

†

a < 3,

†

vf ± are complex. That is, they do not exist as real values between 0 and 1 for

Physics 4550, Fall 2003 – Dynamical Systems 9

†

a < 3. The logistic map has no two-cycle until

†

a > 3. The slope of the tangent line to the curve

†

vn+2

versus

†

vn at the fixed points determines the stability of the fixed points. These slopes are:

†

a 2 at 0,

†

(2 -a )2 at

†

(a -1) a , and

†

-a 2 + 2a + 4 at both

†

vf + and

†

vf - . The magnitude of the latter is less than 1

(and hence the two-cycle is an attractor) up to

†

a = 1+ 6 = 3.449489743... . For larger values of

†

a , thetwo-cycle is no longer stable. The attractors that replace the fixed points and two-cycle can be determinedanalytically by arguments similar to those leading to (6), but the algebra gets pretty scary. Instead, let’slook at numerical results.

The figure to the right shows a superposition of twoplots—one (the parabola) of

†

vn+1 versus

†

vn, the other of

†

vn+2 versus

†

vn. Both plots are for

†

a = 3.5. Note that both curves intersect the45˚ line at 0 and at

†

(a -1) a = 0.71428... . Those are the two fixedpoints of (5). The

†

vn+2 versus

†

vn curve also intersects the 45˚ line at0.857… and at 0.428… . Those are the two-cycle values.

The figures to the rightshow

†

vn+3 versus

†

vn and

†

vn+4

versus

†

vn. The former curve onlyintersects the 45˚ line in two places,at the fixed point values of (5).(Repeat every time also repeatsevery 3 times.) The latter curveintersects the 45˚ line in 8 places,however. These correspond to thetwo fixed points of (5) (repeat every time repeats every 4 times), the two values (6) (repeat every 2 timesalso repeats every 4 times), and 4 more values. These constitute a 4-cycle. This 4-cycle is an attractor for

†

3.449... < a < 3.554... . As

†

a is allowed to increase further, we find that the 4-cycle attractor gives wayto an 8-cycle attractor (at about

†

a = 3.5541... ), which in turn gives way to a 16-cycle attractor (at about

†

a = 3.5644... ), and so on. The periods of successive attractors keep doubling at values of

†

a that getcloser and closer. Eventually, at about

†

a = 3.570... we find that the period of the attractor goes to infinity!That is, the attractor at this value of

†

a never repeats. Such a deterministic, aperiodic behavior is calleddeterministic chaos. The chain of events (at successively higher values

†

a ) preceding the appearance ofchaos is called a period doubling cascade.

It is instructive to make an attractor diagram—a plot of what allthe attractors look like as a function of

†

a . This is shown in the figure tothe right. The horizontal axis is

†

a running from 0 on the left to 4 on theright. The vertical axis is attractor values. Plotted above each value of

†

ais 400 values on the attractor. Where you see just one point above a given

†

a , there are actually 400 identical values. That’s a fixed point. Twopoints is a 2-cycle, and so on. For

†

a above 3.57 we see a mess of points.The mess indicates chaos.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Physics 4550, Fall 2003 – Dynamical Systems 10

Surprisingly, if the mess is magnified, it isn’t as messy as itfirst seems. To the right is shown a portion of the attractor diagramfor

†

3.45 < a < 4 . There’s a big “hole” at about

†

a = 3.83... . Closeexamination shows that this region corresponds to the appearance of astable 3-cycle. As

†

a is varied in this region, we find a cascade ofperiod doublings; 3-cycle to 6-cycle to 12-cycle, etc. See below to theleft. Magnification of central part of this 3-cycle window (belowright) shows a picture that looks very similar to the whole plot. Theattractor diagram of the logistic map is filled with shrunken copies ofitself. It’s a fractal!