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"much more extensive than before.." (The Mathematical Review, March 2004)
"The fully updated second edition provides a selfcontained introduction to the theory and applications of nonlinear dynamics and chaos." (International Journal of Environmental Analytical Chemistry, Vol.84, No.14 – 15, 10 – 20 December 2004)
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Nonlinear Dynamics and Chaos
By J. M. T. Thompson H. B. Stewart
John Wiley & Sons
ISBN: 0471876844Chapter One
An Overview of Nonlinear PhenomenaIn Part I we aim to give a general outline of nonlinear dynamics, which is an essential prerequisite to our more advanced studies including our goal of understanding chaotic motions. This chapter provides a quick overview of the nonlinear dynamics field, before we begin our more detailed presentation.
2.1 UNDAMPED, UNFORCED LINEAR OSCILLATOR
We start our overview by looking at the undamped, unforced linear oscillator of Figure 2.1. The equation chosen for this first illustration has the stiffness constant 4[[pi].sup.2], which makes the periodic time equal to unity. The solution of such an equation is simply a sine wave, the constant amplitude and phase of which are determined by the starting values of x and x. So, once started, we have a constant sine wave that persists for all time, and there is no transient or decay of any kind. The periodic time, unity in the present example, is a constant independent of the starting conditions, the amplitude of the motion, and the time.
A typical plot of x against the time t is shown, resulting from the starting condition (x, x) equal to (1, 0) at the time t = 0. Along with the other, mainly nonlinear, problems considered in this chapter, this solution was obtained by numerical time integration using a fourthorder RungeKutta routine on a desktop HewlettPackard computer with the step size indicated, here [DELTA]t = 0.02.
If we plot not x against t but x against x, we have the phase portrait shown on the left. Starting as before at (1, 0) we now have the closed ellipse shown, the representative point moving continuously round and round this closed orbit as the time goes to infinity. The power spectrum of this response, shown in the top righthand diagram, is simply a spike (or delta function) at the circular frequency of 2[pi] radians per second.
We must finally ask the question: what would happen if we changed the starting condition by a small amount? The answer is illustrated in the lower diagram, where we show both the fundamental reference motion starting at (1, 0) and a perturbed motion starting at (1.02, 2). We see that we have two sine waves running in step with just a small difference in amplitude and phase resulting from the slightly different starting values of x and x. They continue to run nicely in step for all time because the period of oscillation of the two motions is the same (and equal to unity, as we have seen). So a starting perturbation is preserved, and the fundamental motion is neutrally stable in a dynamical sense.
In the lefthand phase space, the two motions appear as neatly nesting ellipses. All possible motions of this linear oscillator are indeed represented by a complete family of nesting ellipses, which represent the full phase portrait of the system. The orbit passing through any particular starting point (x, x) defines the subsequent unique motion of the oscillator.
This linear oscillator models in an approximate fashion many basic physical systems, such as for example the free motions of a simple hanging pendulum. The modelling is however unrealistic in two important ways. First, it ignores the damping action of inevitable dissipative forces, such as air resistance in the example of the laboratory pendulum. In the absence of impressed driving forces, the motions of all real macroscopic mechanical systems will eventually decay, as with a free experimental pendulum, so our present equation fails to model this vital aspect. Secondly, all real systems will have some degree of nonlinearity, which in itself modifies the behaviour in important ways. Largeamplitude oscillations of an undamped pendulum are for example governed by a nonlinear differential equation that we shall examine next: a linear approximation to the behaviour of a pendulum is only valid for small angles of oscillation.
The two unrealistic approximations of linearized stiffness and zero damping will be removed in turn, so we look next at the largeamplitude, nonlinear motions of an undamped pendulum.
2.2 UNDAMPED, UNFORCED NONLINEAR OSCILLATOR
The undamped, unforced nonlinear system of Figure 2.2 represents the exact equation of motion of a simple pendulum undergoing arbitrarily large oscillations. This equation in terms of the angle x is easily derived using Newton's law of motion for the bob by resolving perpendicular to the light string to eliminate the unknown tension: alternatively it can be derived by Lagrangian or Hamiltonian energy methods. The length of the pendulum, relative to the gravitational constant, has been chosen to make the coefficient equal to 4[[pi].sup.2]. So for small oscillations we could linearize the equation by approximating sin x to x, and retrieve the linear oscillator of our earlier discussion, with periodic time equal to unity.
The solution of this nonlinear differential equation can be obtained after some algebra in terms of elliptic integrals: alternatively the equation can be easily integrated numerically on a digital computer as we have done here. Depending on the starting conditions of (x, x) we now find a steady undamped oscillation corresponding to the motion of our idealized undamped pendulum. A given motion from a given start thus exhibits no transient or decay, just a steady waveform of constant amplitude and constant period. The waveform is not however sinusoidal, and could in fact be decomposed by Fourier analysis into a fundamental harmonic plus odd higher harmonics: this gives rise to the power spectrum shown with a large spike at a certain circular frequency [[omega].sub.F] and smaller spikes at 3, 5, 7,... times this value.
The central waveform shows the steady oscillation starting at (3.054, 0) corresponding to the pendulum starting from rest with x = 0 at a value of x = 3.054 x 180/[pi] = 175°. To visualize this physically we must suppose that the heavy pendulum bob is supported not by a string, which could become slack, but by a light rigid rod pivoted to the fixed support. Because this rigidlink pendulum would be in (unstable) equilibrium at x = 180°, the motion begins very slowly and the waveform is very flat and quite noticeably nonsinusoidal. The corresponding x (x) phase picture is shown to the lefthand side: the closed trajectory is quite clearly not elliptical, and has a high curvature on the x axis corresponding to the proximity of an unstable equilibrium state.
Now the periodic time of a given motion is constant as we have just seen, but the period of different motions increases with the amplitude. It is clear for example that a start very close to x = 180° will give a motion with a very large period, since at the end of each big swing the pendulum will almost come to rest in the inverted position: indeed the periodic time goes to infinity as the amplitude approaches [pi]. Notice that the periodic time of our displayed waveform is about 3, compared with the periodic time of unity for the smallamplitude linearized motions.
This variation of period with amplitude gives rise to a new phenomenon when we consider a perturbed motion. The lower diagram shows the fundamental motion just considered together with a perturbed motion starting from slightly different initial conditions. Because these new conditions give rise to a motion with a slightly different amplitude, the perturbed waveform has a slightly different period. So we have a beat phenomenon and the two motions drift in and out of phase with one another. This means that, although the two waveforms will eventually resynchronize, there is an initial divergence from adjacent starts. This makes the fundamental oscillatory motion unstable in the strict sense of Liapunov. In the lefthand phase diagram however, in which the time discrepancies of the two motions are not visible, the two closed orbits are seen to lie everywhere close to one another: in recognition of this fact the fundamental motion is said to be orbitally stable.
For the motions under consideration, the phase portrait of the present undamped nonlinear oscillator consists of nesting closed orbits. For small oscillations these are roughly elliptical corresponding to the nearly sinusoidal waveform, but they become increasingly distorted with increasing curvature near the x axis for the larger nonsinusoidal motions.
The steady undamped oscillations of our first two examples are not typical of real undriven systems. Clearly the smallest trace of dissipation will give damped waveforms, and the nest of closed orbits in the phase space will become inward spirals. The fact that the topological nature (closure) of the phase orbits can be destroyed by even infinitesimal damping is recognized by declaring the pathological undamped systems to be structurally unstable.
For the rest of this chapter we shall be concerned with typical damped systems, and we start by looking at the behaviour of a damped linear system.
2.3 DAMPED, UNFORCED LINEAR OSCILLATOR
We consider then the differential equation of Figure 2.3, which is written in a rather standard form, with [zeta] representing the damping factor, namely the ratio of the actual damping to the critical damping at which oscillatory behaviour ceases. We can think of this equation as representing the motion of a mass constrained by a linear elastic spring in parallel with a dashpot full of oil, which is assumed to provide a force opposing the instantaneous velocity.
An analytical solution of this linear differential equation is readily written down: for light damping with [zeta] < 1 we have an exponentially damped sine wave, while for heavy damping with [zeta] > 1 we have a nonoscillatory exponential decay.
A typical lightly damped waveform is shown in the middle picture, starting at x = 1, x = 0. The decaying wave has a constant period, defined for example by successive crossings of the time axis, which is nevertheless slightly dependent on the value of [zeta]. With the light damping shown, the period is essentially unchanged from the period, 2[pi], of the corresponding undamped system obtained by setting [zeta] = 0. For light damping the power spectrum will be roughly a single spike decaying to zero along with the wave amplitude.
The corresponding phase portrait on the left is now a spiral, heading inwards towards the asymptotically stable equilibrium state at the origin (0, 0). The full linear phase portrait, termed a focus, is a set of intertwining, noncrossing spirals. Every motion here represents a transient to the asymptotically stable equilibrium state of rest at the origin, which for obvious reasons is called a point attractor. The whole phase portrait is now structurally stable since for finite damping the spiralling form cannot be topologically changed by any infinitesimal changes to the system.
The pictures for heavier supercritical damping shown below give the waveform and phase trajectories for six alternative starts. The system moves back to its stable state of rest in a direct nonoscillatory fashion, and the whole phase portrait is called a node. Once again, we have a structurally stable point attractor at (0, 0) capturing all motions of the system.
Since all motions decay to rest, fundamental and perturbed motions coalesce as time goes to infinity, and starting perturbations are lost.
2.4 DAMPED, UNFORCED NONLINEAR OSCILLATOR
To conclude our examination of unforced (undriven) systems, we look now at a damped nonlinear problem, typified by the pendulum of Figure 2.4. This is the largeamplitude pendulum of our earlier discussion, now with the modelling of air drag by a realistic velocitysquared law: notice that the damping force proportional to chi square has to be entered into the differential equation of motion as xx to ensure that it is always opposing the velocity. Having put on this quadratic damping, we should perhaps emphasize that the form of damping is largely irrelevant to the following discussion, the salient points being just as well illustrated by the use of linear damping: the computed traces relate however to the quadratic damping.
Clearly we once again have transients to the asymptotically stable hanging equilibrium state representing a point attractor in the phase space.
The central waveform damps and becomes increasingly sinusoidal as x becomes small, while the power spectrum is a decaying set of spikes as shown. The phase portrait is a spiral, becoming increasingly elliptical as the trajectories approach the central attractor. A little linear damping would be needed to make this portrait structurally stable near the origin.
As with the undamped pendulum suffering largeamplitude oscillations, adjacent starts still exhibit a temporary beating character with an associated initial divergence due to the variation of the period with amplitude. But initial perturbations are eventually lost as all motions coalesce in the unique hanging state.
This local phase portrait is a set of intertwining spirals with all motions captured by the central attractor. The full phase portrait of a pendulum including highvelocity motions passing through the inverted state is most nicely seen in a cylindrical phase space, and will be presented later (Figure 3.15).
2.5 FORCED LINEAR OSCILLATOR
We have so far looked only at autonomous unforced systems with zero on the righthand side of the equation, but we turn now to sinusoidally driven nonautonomous oscillators. Damping, we have seen, is an essential ingredient of good modelling, so we shall start by looking at the damped, forced linear oscillator of Figure 2.5. This would be an adequate mathematical model of a pinended steel beam driven to smallamplitude lateral oscillations by an electromagnet carrying a sinusoidal alternating current. Here physical damping would arise from air resistance and internal material dissipation. The numerical coefficients have been chosen to provide a sharp frequency contrast between the transient and the steadystate solution, and the damping ratio of the unforced lefthand side is 0.1 (see equation 3.11).
This is a classical resonance problem of engineering texts, and the wellknown analytical solution is easily written down. It is the algebraic sum of the socalled particular integral (PI) and the complementary function (CF).
Continues...
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