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Chaotic Dynamics of Nonlinear Systems
By S. Neil Rasband Dover Publications, Inc.
Copyright © 1990 S. Neil Rasband
All rights reserved.
ISBN: 978-0-486-80577-1
CHAPTER 1
INTRODUCTION
There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.
(W. Shakespeare, Hamlet, Act I, Scene 5)
Arguably the most broad based revolution in the worldview of science in the twentieth century will be associated with chaotic dynamics. Yes, I know about Quantum Mechanics and Relativity, and for physicists and philosophers these theories must rank above Chaos for their impact on the way we view the world. My assertion, however, refers to science in general, not just to physics. Leaving improved diagnostic instrumentation aside, it is not clear that Quantum Mechanics or Relativity have had any appreciable effect whatever on medicine, biology, or geology. Yet chaotic dynamics is having an important impact in all of these fields, as well as many others, including chemistry and physics.
Surely part of the reason for this broad application is that chaotic dynamics is not something that is part of a specific physical model, limited in its application to one small area of science. But rather chaotic dynamics is a consequence of mathematics itself and hence appears in a broad range of physical systems. Thus, although the mathematical representations of these physical systems can be very different, they often share common properties. In this introductory chapter we outline in a qualitative way some of the common features of chaos and introduce the reader to some chaotic phenomena. We further introduce some of the methods employed in the study of chaotic dynamics. Precision is left to discussions in subsequent chapters.
1.1 Chaos and Nonlinearity
The very use of the word "chaos" implies some observation of a system, perhaps through some measurement, and that these observations or measurements vary unpredictably. We often say observations are chaotic when there is no discernable regularity or order. We may refer to spatial patterns as chaotic if they appear to have less symmetry than other, more ordered states. In more technical terms we would say that the correlation in observations separated by either space or time appears to be limited. However, from the outset we must make clear that we are not speaking of the observation of random events, such as the flipping of a coin. Chaotic dynamics refers to deterministic development with chaotic outcome. Another way to say this is that from moment to moment the system is evolving in a deterministic way, i.e., the current state of a system depends on the previous state in a rigidly determined way. This is in contrast to a random system where the present observation has no causal connection to the previous one. The outcome of one coin toss does not depend in any way on the previous one. A system exhibiting chaotic dynamics evolves in a deterministic way, but measurements made on the system do not allow the prediction of the state of the system even moderately far into the future.
Whenever dynamical chaos is found, it is accompanied by nonlinearity. Nonlinearity in a system simply means that the measured values of the properties of a system in a later state depend in a complicated way on the measured values in an earlier state. By complicated we mean something other than just proportional to, differing by a constant, or some combination of these two. Although by these remarks, we do not mean to imply that somewhat complicated phenomena cannot be modeled by linear relations.
A simple, nonlinear, mathematical example would be for the observable x in the (n + 1)th state to depend on the square of the observable x in the nth state, i.e., xn+1 = x2n. Such relations are termed mappings, and this is a simple example of a nonlinear map of the nth state to the (n + 1)th state. A familiar physical example would be the temperature from one moment to the next as water is brought to a boil. At the end of this process the temperature in the (n + 1)th state is just equal to the temperature in the nth state, but this is clearly not true as the water is being heated to its boiling temperature. Frequently the problem of modeling real-world systems with mathematical equations begins with a linear model. But when finer details or more accurate results are desired, additional nonlinear terms must be added.
Naturally, an uncountable variety of nonlinear relations is possible, depending perhaps on a multitude of parameters. These nonlinear relations are frequently encountered in the form of difference equations, mappings, differential equations, partial differential equations, integral equations, or even sometimes combinations of these. As we look deeper into specific causes of chaos, we shall see that chaos is not possible without nonlinearity. Nonlinear relations are not sufficient for chaos, but some form of nonlinearity is necessary for chaotic dynamics.
Having considered briefly nonlinear mappings, we now consider somewhat more closely systems modeled by differential equations. It is convenient when discussing the properties of differential equations to write them in a standard, first-order form:
[??] = f(x, t). (1.1)
If the f in (1.1) is independent of t, then the equation is said to be autonomous; otherwise it is nonautonomous. For such a system to be chaotic it must have than one degree of freedom, or be nonautonomous. We illustrate this with the familiar example of a simple pendulum. The differential equation for a simple pendulum is often written in the form
[??] + w20 sin x = 0, (1.2)
where x represents the angular displacement of the pendulum from the vertical position, two overdots denote two derivatives with respect to time in the usual way, and w0 denotes the natural frequency of the pendulum for small angular displacements. Even though this system is highly nonlinear, it does not exhibit chaotic dynamics. There is only the single degree of freedom associated with x and the right-hand side is the constant zero. If, instead, we replaced the zero in (1.2) with some function f(x, t), then the system becomes nonautonomous and may exhibit chaotic dynamics, depending of course, on the exact nature of the function f(x, t). In effect the time t has become an additional degree of freedom.
To put the differential equation (1.2) in the standard form (1.1) and to make explicit the notion that time is a degree of freedom, we define a new independent variable θ, and a new dependent variable y = dx/dθ. Then with the driving term f(x, t) on the right, (1.2) becomes the system
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
In this form the system consists of three, first-order differential equations and is nonautonomous. Frequently, such a system is said to have 1 1/2 degrees of freedom, since very often dynamical systems, particularly those resulting from Hamiltonian mechanics, have a pair of equations for every degree of freedom.
Although simple quadratic maps and forced, nonlinear oscillators like the preceeding examples may not appear to offer much promise for displaying a rich diversity of chaos, the opposite is true. We will see that indeed within these very simple nonlinearities lurk the seed of nearly all chaotic phenomena, and the bulk of this work is devoted to the study of such simple systems.
One of our major objectives is to classify and characterize deterministic systems exhibiting chaotic dynamics. Thus our characterization of nonlinearity as an essential ingredient for chaotic dynamics marks the beginning of this classification effort. We have further pointed out that for a system with one degree of freedom the differential equation must be nonautonomous. We now illustrate these points and the development of chaos with the familiar example of a simple harmonic oscillator.
1.2 The Kicked Harmonic Oscillator
To introduce many of the concepts and ideas that will be studied in subsequent chapters, we study the motion of a simple harmonic oscillator subject to a periodic impulse. We refer to this system as the kicked harmonic oscillator. The equation of this system is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.3)
where w0 is the natural frequency of the oscillator, A is the amplitude of the kicks, and f(x) is an arbitrary function of x, but not of t. Figure 1.1 shows the familiar phase-plane trajectories for the case where A = 0, i.e., the harmonic oscillator without kicks. Each ellipse corresponds to a fixed value of the energy of the oscillator. With A ≠ 0, the right-hand side of (1.3) depends on time t; this differential equation is therefore nonautonomous.
In an interval between kicks the right-hand side of (1.3) is zero, and the solution is familiar:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.4)
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.5)
where k = 1, 2,. ... For each k, at t = kT we demand that the position of the one-dimensional oscillator be continuous but that the velocity (momentum) change discontinuously. This discontinuous change in the velocity is computed by integrating (1.3) from (kT – ε) to (kT + ε) and then taking the limit as ε -> 0. We find easily the following relationship between the coefficients in the k interval and those in the (k + 1) interval.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.7)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.8)
The subscript k on x and [??] refer to a time infinitesimally prior to the kick at kT. Using (1.8) with (1.6) and (1.7), plus a little algebra, yields the relation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.9)
which gives the position and velocity just before the time (k + 1)T in terms of the position and velocity just before the time kT.
The relationship between the coefficients in the k interval to those in the (k + 1) interval is an example of a two-dimensional mapping. Choosing the driving term in (1.3) to be a sum of delta functions is the feature that allows us to obtain the solution to the differential equation for the kicked harmonic oscillator in terms of the mapping represented by (1.9). The nonlinearities are introduced by the choice made for f(x). With A and f(x) not equal to zero, the system is nonautonomous and thus equivalent to more than one degree of freedom.
For f(x) = 1 or x, the mapping (1.9) is linear and invertible. In light of our previous remarks, no chaotic dynamics is to be expected. Such a case is, however, still nonautonomous — just not nonlinear. A plot of a segment of a phase-space trajectory for f(x) = 1 is given in Fig. 1.2. The trajectory crossings are a consequence of the time dependent driving term but can be eliminated by plotting the trajectory in extended phase space as in Fig. 1.3.
From (1.6) and (1.7) with f = 1 we obtain immediately
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.10)
with k = 2, 3, ... and Ω = 2π/T. If ω0 = Ω, i.e., if the kicks come at a frequency equal to the natural frequency of the oscillator, the coefficient Bk -> ∞ with k. The velocity and hence the energy of the oscillator become unbounded. This situation is called resonance. Resonance is a phenomenon occurring in a great many nonlinear systems leading to the destruction of the integrable behavior. The issue of resonance will reappear often in subsequent sections as we consider dynamics of nonlinear systems.
Forω0 ≠ Ω the series in (1.10) can be summed to give
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.11)
If the ratio (ω0/Ω) is a rational number, then there will always exist some k for which Ak and Bk return to their inital values, and the system is periodic.
As an alternative to a trajectory plot in extended phase space, which becomes impractical after a few periods, it is convenient to study the time evolution of this system by making a point in the (x, [??]) phase plane at the values of t = T, 2T, ..., i.e., at values of t corresponding to multiples of the period of the driving function. Such a plot for a dynamic system is called a Poincaré section. Figure 1.4 is a Poincaré section plot for the system represented by (1.10), (1.11) and we see that the phase points always lie on ellipses, just as for the oscillator without kicks.
Comparing an orbit in Fig. 1.1 with the orbit in Fig. 1.2 dramatically demonstrates that a linear, time-dependent driving term alters the orbits in phase space. But this change in the nature of the phase-space orbits still does not go so far as to produce any chaotic dynamics. The relation between the (Ak, Bk) and (Ak+1, Bk+1) is still linear in (1.10) and (1.11). Nonlinearity is still absent in the system producing Fig. 1.4. Exercise 1.3 considers the same issues with f(x) = x.
We now change from f(x) = 1 to f(x) = x4 and examine the Poincaré section plots for orbits with initial conditions similar to those producing the plots of Fig. 1.4. The Poincaré sections now produce Fig. 1.5, which is quite different from Fig. 1.4. We see two highly distorted elliptical orbits, an inner and an outer one, enclosing a seven-period island chain. Around the outer edge of this island chain there is a small, but finite, layer of chaotic orbits. The centers of the islands are called O points and the points between, joining the individual "islands," are called hyperbolic or X points. The insert in the center of Fig. 1.5 shows a magnified view of the intersection points of a single orbit in the neighborhood of the indicated X point. The reader should bear in mind that the insert only shows one of the seven X points, all of which are connected by a thin chaotic layer around the island structure. The chaotic region occupies a small but finite region in the phase plane. One of the most characteristic features of chaotic dynamics can be seen by considering two trajectories in the chaotic region that have nearly identical initial conditions. After a finite number of iterations of the map, the intersection point for one trajectory is completely unrelated to the intersection point for the second trajectory. This is our first example of chaotic behavior from deterministic dynamics. This feature is commonly referred to as sensitive dependence on initial conditions. Despite sensitive dependence on initial conditions and numerical roundoff, Hammel et al. (1988) have shown that the computation of chaotic orbits for a large number of periods as in Fig. 1.5 is still meaningful.
These few examples, and the kicked harmonic oscillator in particular, have illustrated the necessity for nonlinearity in producing chaotic dynamics. We further illustrated how Poincaré sections can be a useful tool in displaying chaotic consequences. For the kicked harmonic oscillator it was possible to obtain a mapping to advance the system in time, and it should be clear that this is much easier than the numerical integration of a system of differential equations. Partly because maps are easier to advance, and partly because of the importance of Poincaré section maps, considerable attention is devoted to mappings in subsequent chapters. This begins in the next chapter with a study of one-dimensional maps where we also develop additional methods to supplement Poincaré plots for studying and recognizing chaotic behavior.
(Continues...)
Excerpted from Chaotic Dynamics of Nonlinear Systems by S. Neil Rasband. Copyright © 1990 S. Neil Rasband. Excerpted by permission of Dover Publications, Inc..
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