Basics of Nonlinear and Chaotic Dynamics
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In this introductory Chapter we develop the basis of nonlinear dynamics and chaos theory to be used in the subsequent Chapters. After a basic introduction into attractors and deterministic chaos, a brief history of chaos theory is given. Then, temporal chaotic dynamics is developed, both in its continuous form of nonlinear ordinary differential equations (ODEs) and in its discrete form (nonlinear iteration maps). Spatio–temporal chaotic dynamics of nonlinear partial differential equations (PDEs) follows with some physiological examples. The Chapter ends with modern techniques of chaos–control, both temporal and spatio–temporal.
1.1 Introduction to Chaos Theory Recall that a popular scientific term deterministic chaos depicts an irregular and unpredictable time evolution of many (simple) deterministic dynamical systems, characterized by nonlinear coupling of its variables (see, e.g., [GOY87, YAS96, BG96, Str94]). Given an initial condition, the dynamic equation determines the dynamic process, i.e., every step in the evolution. However, the initial condition, when magnified, reveals a cluster of values within a certain error bound. For a regular dynamic system, processes issuing from the cluster are bundled together, and the bundle constitutes a predictable process with an error bound similar to that of the initial condition. In a chaotic dynamic system, processes issuing from the cluster diverge from each other exponentially, and after a while the error becomes so large that the dynamic equation losses its predictive power (see Figure 1.1). For example, in a pinball game, any two trajectories that start out very close to each other separate exponentially with time, and in a finite (and in practice, a very small) number of bounces their separation δx(t) attains the magnitude of L, the characteristic linear extent of the whole system. This property of sensitivity to initial conditions can be quantified as
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1 Basics of Nonlinear and Chaotic Dynamics
Fig. 1.1. Regular v.s. chaotic process.
|δx(t)| ≈ eλt |δx(0)|, where λ, the mean rate of separation of trajectories of the system, is called the Lyapunov exponent. For any finite accuracy |δx(0)| = δx of the initial data, the dynamics is predictable only up to a finite Lyapunov time 1 TLyap ≈ − ln |δx/L|, λ despite the deterministic and infallible simple laws that rule the pinball motion. However, a positive Lyapunov exponent does not in itself lead to chaos (see [CAM05]). One could try to play 1– or 2–disk pinball game, but it would not be much of a game; trajectories would only separate, never to meet again. What is also needed is mixing, the coming together again and again of trajectories. While locally the nearby trajectories separate, the interesting dynamics is confined to a globally finite region of the phase–space and thus the separated trajectories are necessarily folded back and can re–approach each other arbitrarily closely, infinitely many times. For the case at hand there are 2n topologically distinct n bounce trajectories that originate from a given disk. More gen
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