Classical Mechanics notes (10 of 10)

7/30/2019 Classical Mechanics notes (10 of 10)

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10Chaos in the Damped Driven Pendulum

(Most of the material in this chapter is taken from Thornton and Marion, Chap. 4)

Introduction

We looked at the damped driven simple harmonic oscillator in Chapter 2. We saw thatthe solution to the equations of motion could be found in a simple analytic form.

However, the damped driven pendulum has a different equation of motion, because it is a

non-linearoscillator under large amplitude swings. That is, we cant use the

approximation sin when is large. This seemingly small change has a dramatic

effect on the overall dynamics.

2

0

20

2 cos Damped Driven Simple Harmonic Oscillator

2 sin cos Damped Driven Pendulum

A t

A t

+ + =

+ + =

(0.1)

The Damped Driven Pendulum (DDP) has no analytic solution. That is, there is no

known solution to the equations of motion that can be expressed as a finite series of

polynomials and trigonometric functions. Not only is there no known solution, but it canbe rigorously shown that no finite analytic solution exists! By no means is that equation

the only one that lacks an analytic solution: many more are known, including the famous

three-body problem of celestial mechanics (i.e. the motion of three bodies moving only

under their own gravity).

These systems are found to have some interesting properties, categorized as chaotic,

that well investigate in the context of the DDP. Note that the word chaotic and chaosin physics does not imply randomness in the usual sense. Chaotic systems like the DDP

are completely deterministic: knowledge of the initial conditions allows one to solve

unambiguously for the future motion at all times. However, chaotic systems often appearrandom because of chaos trademark sensitivity to small changes in the initial conditions,

which well investigate further later.

Numerical investigations of chaos

The absence of an analytic solution to the DDP (and other chaotic systems) means that

one of the few practical ways to study its behaviour is by numerical means. In thistechnique, one plugs in the initial conditions into the equations, and integrates them

numerically over short time steps t. For example, given the initial position and angular

velocity of the pendulum and , one can approximate by

0 0

2

0 0

( 0) ; ( 0)

( 0) 2 ( 0) sin ( 0) cos (0)

t t

t t t A

= = = =

= = = +

(0.2)


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Now that one has , one can compute an approximate value for and a small time t

in the future by simply assuming and are constant over this small time step

( ) ( 0) ( 0) ;

( ) ( 0) ( 0)

t t t t t

t t t t t

= = + =

= = + =

(0.3)

Since one now has and at our new time t= t, we can go on and simply repeat this

process to get the values at t= 2t,by calculating ( )t t =

2

0 0( ) 2 ( ) sin ( ) cos ( )t t t t t t A t = = = + (0.4)

and going on to calculate and at t= 2t.

( 2 ) ( ) ( ) ;

( 2 ) ( ) ( )

t t t t t t t

t t t t t t t

= = + =

= = + =

(0.5)

and continuing on we can integrate the equations of motion indefinitely into the future

2

0 0( 2 ) 2 ( 2 ) sin ( 2 ) cos (2 ),

,

t t t t t t A t

etc etc

= = = + (0.6)

A valid question is whether or not the approximations we have made above are valid.

Usually, if the time step is small we are OK. Also, we can use more sophisticated

methods of numerical integration (which we wont get into here). Overall, the question of

error in the numerical integration itself can be problematic, but we wont get into thisissue here.

Phase plots

How can we represent the results of our numerical integrations? What we get out is a

sequence of positions and velocities and over time. Since this is enough information

to specify almost all we need to know about the particle, this is in some sense the solution

we want (Like most dynamical systems, knowing the position and velocity is enough tospecify all future motion, though this doesnt necessarily mean that any accelerations or

higher derivatives of the position are not potentially of interest). So a simple way to

display the solution produced is a phase plot, where the phase space (which in this case

has only 2 dimensions and is plotted). Note that the system has only one degree of

freedom, represented by but we need to completely understand the motion (actually,the phase of the driving term is also important, and well return to that).


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Figure 1: the phase plot of a simple undampled pendulum under small oscillation

Consider a simple undamped pendulum. If is near zero, the motion is approximately

that of a simple harmonic oscillator. Thus and are described by sin or cos, and are

90 degrees out of phase (e.g. is a maximum when 0= , and is a maximum

when 0= ). A result, the phase plot looks like circles (or ellipses, depending on exactly

what units you use for the axes) around the point 0 = = .

Figure 2: phase plot of a damped undriven pendulum


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In the case of a damped pendulum that is not driven, the amplitude decays down over

time, and so the phase trajectory would spiral into the point 0 = = over time. It will do

so faster if theres a lot of damping, slower if theres less.

The phase plot of a real pendulum must include the possibility of very large oscillations;

in fact it must include the possibility of rotations all the way around, clockwise orcounterclockwise, if the conditions are right. A phase plot which shows the varied

possible motions for a simple undamped pendulum (SUP) is shown below.

The motion near the point 0 = = corresponds to small amplitude oscillations. As the

amplitude gets larger, the circles (ellipses) get distorted in shape because the pendulum is

not a linear oscillator. Nevertheless we continue to get stable periodic oscillations up until

the point where the amplitude is such that the pendulum can reach = (that is, straightup, with the mass above the pivot point). If the amplitude (again, for a SUP) is large

enough to take it over the top, the pendulum will continue to go around and around in

the same direction indefinitely: there is no more oscillation, but instead motion that we

call rotation. The boundary line between the two types of motion is called theseparatrix, and represents the phase plot of a pendulum with the minimum energy for the

bob to go all the way around. The point at 0 = = is called a stable equilibrium,because a pendulum initially located there, if subject to a small perturbation, will remain

in the vicinity of this point in phase space (the pendulum will perform small oscillations,

but in phase space this corresponds to small circles around the point 0 = = ). The

point ; 0 = = is a point ofunstable equilibrium because though it is an equilibrium

point, a pendulum placed there, if slightly perturbed, will move quickly away from this

point in phase space.

Figure 3: Phase plot for undamped simple pendulum of different energies


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For a SUP, the phase plots are easy to determine and unchanging. Note that each phase

curve is closed, the motion is exactly periodic. Note that for an SUP, once the phasecurve closes, the motion must continue to repeat itself forever in the same way again.

This is because, since the future motion is completely specified by and , if these

return to their initial conditions, the future motion must repeat.

For the DDP, things get more complicated. Energy is being removed by the damping and

put in by the forcing, so one cant simply assign a constant energy to a pendulum and

thus determine its future motion. In fact, the phase plots of DDP can be quite complex,but we can consider them against a background of our simple undamped pendulum to

better understand them.

Numerical simulations

We will perform a few numerical simulations of the DDP to better understand its motion.

These will be done here with the Non-Linear Oscillations package by PhysicsAcademic Software (free) http://faculty.ifmo.ru/butikov/Nonlinear/index.html

Figure 4: Screen capture of Non-Linear Oscillations (Demo example: Low

Frequency Small Oscillations)


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Figures 4 and 5 show the screen captures for small and large oscillatory motions for the

same DDP (same damping, same driving, but different initial conditions). The first is thestandard small oscillations we have discussed. The second shows much larger

oscillations, but ones that will be maintained indefinitely given the driving. Note that the

phase plot (upper left hand corner of both screen captures) is rather circular for the smalloscillations, and rather distorted for the second figure, as we have motion which is ratherclose to the separatrix in this case.

Figure 5: Demo example: Low Frequency Large Oscillations

From the smooth repeating results of the two simulations above, we might expect that the

system is not undergoing chaotic behaviour. Though this is in fact the case, we might ask,

how can we tell if chaos is occurring? Certainly dramatic chaos is in some ways obvious(as well see) but what exactly are we looking for? For example, is Figure 6

representative of chaotic motion?

Perhaps you suspect chaos because the pendulum sometimes goes over the top, and

sometimes doesnt (This can be seen in the plot of angle vs time, or from the way that

the phase plot crosses the separatrix (shown in red)). However, this is not unequivocal

evidence for chaos. Perhaps you suspect chaos is at work because the phase plot crosses


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over itself many times, which does not occur for the SUP, or the two earlier examples of

the DDP. However, the repetition of the same values of and does NOT ensure that

the future motion will be exactly the same. Why? Because there is a forcing term as well.

If the forcing terms is also at the same phase so that and and the forcing return to

their initial conditions, then the motion will repeat. Then we have a periodic solution.

It turns out that non-chaotic motion in the DDP can be represented by a finite number of

sine waves, of varying amplitudes and frequencies, while chaotic motion cannot. How

can we determine from our phase plot if this is the case? The key difference here is that

non-chaotic motion will be periodic, that is, at some point , and the forcing term t

will return to exactly the same values. How can we determine this, since plotting all three

would require plotting in three dimensions (a little tricky) and the constant plotting of thephase trajectory itself can complicate the plot?

Figure 6: Chaotic motion or not?

Instead of plotting the whole phase trajectory, lets just plot its value when the forcing

term tis at certain fixed values (eg t=0 ort=, or whatever). So we just plot one doton our phase plot every cycle of the forcing function. If the motion is periodic, then the

dots will have to eventually start plotting on top of each other and there will be a finite


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number of dots. If the motion is not periodic, the dots in this Poincare section will never

repeat: this is an indicator of chaos.

The upper left hand portion of Figure 6 is reproduced in Figure 7. The phase trajectory is

in blue, and the white circles are the points of Poincare section. Though the motion is

plotted over a very long time, there are only nine dots in the Poincare section, so themotion must be repeating itself in a periodic fashion. Thus the motion of Figure 6 is notchaotic, but periodic, repeating every nine cycles of the forcing function.

Figure 7: Blow-up of phase plot from Figure 6

A complete understanding of Poincare sections would take more time than we have here,

but suffice it to say that if the Poincare section of the DDP consists of a finite number of

points or points which all fall on a single one-dimensional curve on the phase plot, the

motion is not chaotic. If not, it is. Lets actually take a look at some chaotic motion.

Chaotic motion

Chaotic motion of the DDP is shown in Figure 8. The horizontal and vertical traces havegone around several times, and there doesnt seem to be an obvious pattern. But what

does the Poincare section show? A blow-up is in Figure 9. You can see that the dots of

the Poincare section do not seem to be repeating themselves. But what if the motion justhas a very long period? In principle, you have to wait a long time to find out if the motion

repeats, but the chaotic nature of the DDP usually reveals itself within a reasonable

calculation time. In fact, the motion shown in the figures is chaotic. This can be more


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easily seen from a higher-resolution Poincare section of the motion with the package

Chaos demonstrations http://www.webassign.net/pas/chaos_demo/chaos_demo.html

that has a free demo on the web.

Figure 8: Chaotic motion in the DDP

Figure 9: Phase plot from Figure 8


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Figure 10: High-resolution Poincare section for chaotic motion in the DDP

Figure 10 shows the Poincare section for a DDP under chaotic conditions. Note that there

is not a finite number of points nor are the points confined to a one-dimensional curve inthe plot. The points fall only in particular parts of the plot, in a way which clearly has a

lot of order. In fact, zooming in on one particular part of the plot reveals finer and finer

levels of detail. The Poincare section of a chaotic system is in fact a fractal, a beautiful

mathematical entity associated with chaos. It derives its name from the fact that it is in amathematical sense of fractional dimension (the Poincare section is somewhere between

one dimensional and two dimensional).

What is chaotic motion?

OK, so weve made some progress in identifying chaotic motion in the DDP. But what is

chaotic motion really? This question is not easy to answer completely: chaos is still undergoing a lot of study. But we can identify some hallmarks and characteristics of chaos to

try to understand it better.

Chaos identification

What clues can we look for to try to identify chaos? We might look for the absence of ananalytic solution, but this is not a sufficient condition. Weve seen that many solutions to

the DDP are periodic (not chaotic) though we dont have an analytic solution. In addition

to the Poincare section, there are other ways to identify chaos.


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One way which weve already touched on is the idea that non-chaotic solution can be

expressed as a finite number of analytic functions. As a result, if you take the Fourier

transform of say, as a function of time, only a finite number of frequencies appear.Chaotic motion is characterized by the presence of (in principle) an infinite number offrequencies, though youd have to take the Fourier transform of an extremely long data

set ofversus time to get them all. Still a Fourier transform can be a useful clue to theexistence of chaos.

One defining characteristic of chaos is that very similar initial conditions produce

behaviour that is (after a certain amount of time) very far apart in phase space. Consider

two small twigs placed side by side in a turbulent stream. For a certain amount of time,the twigs will remain near each other but eventually they will separate and go along very

different paths. This is also true of the DDP. Two initial conditions which are even an

arbitrarily small distance apart in phase space will eventually diverge to the point whereone wouldnt guess that they actually started near each other. This behaviour is different

from non-chaotic systems. Consider two SUP that are started with slightly different initial

and/or

. Their motions, while different, will deviate from each other only slowly, ornot at all. Non-chaotic systems can be shown mathematically to usually diverge linearly

with time, while chaotic systems diverge exponentially.

This rapid divergence of two very similar initial conditions is part of what makes chaotic

systems so difficult to predict. The systems are completely deterministic and so one

should be able to use the initial conditions to compute all future behaviour (and one can).

However, unless one knows the initial conditions to infinite precision (impossible inpractice), one cannot compute the true motion of the system beyond a certain point.

Consider the motion of the planets in the Solar system, which is mildly chaotic on million

year time scales. Any prediction of where the planets were or will be in the distant future

or past is impossible to make, unless we know their current positions and velocitiesperfectly. Note that thanks to exponential divergence, knowing their positions to within

nanometers only buys a little extra time over knowing them to within meters orkilometers. As a result, even if you knew all the equations of physics and the positions

and velocities of every particle in the Universe, you could still not predict the future

behaviour of the Universe unless you knew the positions and velocities of all the particles

perfectly. A tall order

Weve mentioned that chaotic systems are characterized by exponential divergence in

phase space. A couple of points 1) since the motion is often bounded (doesnt go off toinfinity), chaotic systems often fold back on themselves in complicated ways (we wont

explore this further here) and 2) the exponential divergence is only on average, but stillcan be looked for by computing the Lyapunov exponents.

Every dynamical system (chaotic or otherwise) has as many Lyapunov exponents as it

has variables. What is a Lyapunov exponent? Consider for example, the difference

between the position of two pendula with very similar initial conditions at two

subsequent times t=0 and t=t. A rough and ready approach is to simply plot thedifference over time and see if its exponential or not. More usually, one plots the ln of


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the difference, suitably normalized from one step to the next. If the motion is diverging

on average slower than an exponential, then the ln function will reduce the average to

zero over time. If not, then the difference will grow linearly with time. Consider if thesystem is chaotic then

t

e

(0.7)

But by taking the ln we reduce it to

ln ln( )te t (0.8)

where is our Lyapunov exponent (what we are trying to measure). If the Lyapunov

exponent is zero or less, the motion is not chaotic. If it is, the Lyapunov exponent gives

us an idea of how far into the future (or past) we can expect to predict the motion of thesystem if we have the initial conditions. Since the difference between the true motion of

the system and the one we would calculate from our initial conditions will diverge likete , a rough measure of how long we can expect to predict the motion is the amount oftime it takes the trajectories to diverge by a factor ofe. This is just given by

1 or 1/t t = = . So we can expect to predict the motion over a time equal to about

1/, which is called the Lyapunov time. So as a rule of thumb, we can only calculate themotion of a system for a few Lyapunov times into the past or future before our

predictions become so far from the true motion that they are effectively useless.

A few notes: a more careful analysis will show that the better you know the initial

conditions, the more Lyapunov times you can expect to make your predictions. But

because of the exponential divergence, you need radical improvements to your initial

conditions to get only a small increase the length of valid predictions. Consider theweather: the Earths atmosphere has a Lyapunov time of only a few days. That is why,

despite ever increasing accuracy in weather data collection, weather forecasts are only

good for a week at best. This is sometimes referred to as the Butterfly effect because ahypothetical butterfly flapping its wings (or any small unmeasured effect on the

atmosphere) somewhere in the world creates a change in the atmosphere that will result

in long-term predictions being incorrect.

The source of chaos

If chaos is tricky to define, can we ask what its source is? What have we added to theDDP that makes it chaotic, while a SUP is not? Thats not quite the right question to ask,

because a DDP can be both chaotic and periodic, depending on the parameters of the

damping, forcing and initial conditions. Is there something we can say about these

parameters that help us identify the source of chaos? The answer is actually yes, thoughwe will only consider the question in a qualitative way.


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The answer is that the motion may be chaotic when the parameters are such that the

pendulum gets near the unstable equilibrium point ; 0 = = in phase space. If the

motion of the pendulum is such that it reaches the vicinity of an unstable equilibriumpoint (and the separatrix), chaos can be expected.

The unstable equilibrium point at ; 0 = =

corresponds to the pendulum straight upwith no angular velocity. If the pendulum gets near this state, then a very small change in

its motion caused by damping or forcing will make all the difference between whether or

not the motion goes over the top (rotation) or swings back down (oscillation). In fact

it is at this point that the exponential chaotic quality of the divergence of two nearlyidentically set-up systems appears (recall the exponential divergence is only an average).

Most of the rest of the time, two nearly identical set-ups will actually diverge much more

slowly than exponential.

The chaos arises in some sense because two initially very similar trajectories can undergo

subsequently very different motion if they cross the separatrix. The presence of an

unstable equilibrium point (which can be shown to always have a separatrix runningthrough it) along with a system that can reach the vicinity of the separatrix in phase space

are necessary ingredients for chaos.

Figure 11: Possible motion of the DDP near the unstable equilibrium point

Classical Mechanics notes (10 of 10)

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