Systems and Control
Broad in scope, this text shows the multidisciplinary role of dynamics and control, presents neural networks, fuzzy systems, and genetic algorithms, and provides a self-contained introduction to chaotic systems.
TABLE OF CONTENTS Systems and Control
Chapter 11: Chaotic Systems and Fractals
An apparent paradox is that chaos is deterministic, generated by fixed rules which do not themselves involve any elements of change. We even speak of deterministic chaos. In principle, the future is completely determined by the past; but in practice small uncertainties, much like minute errors of measurement which enter into calculations, are amplified, with the effect that even though the behavior is predictable in the short term, it is unpredictable over the long run.
Chaos and Fractals: New Frontiers of Science [227, p. 11]
11.1 Chaotic Systems Are Dynamical Systems with Wild Behavior
The objective of this chapter is to show the reader "the wild things that simple nonlinear equations can do" [198, p. 459]. These simple nonlinear models with very complicated dynamics are examples of chaotic systems, which are deterministic nonlinear dynamical systems that exhibit a random-like behavior. A discrete-time model of population dynamics discussed in the next section, Newton's method for finding roots of certain polynomials, or a jetting balloon after its release are examples of dynamical chaotic systems. Trajectories of chaotic dynamical systems are sensitive to initial conditions in the sense that starting from slightly different initial conditions the trajectories diverge exponentially. Another property of a chaotic dynamical system is loss of information about initial conditions. We follow Moon [209, p. 4] to explain this property. Suppose that we are given a state-plane portrait of a chaotic dynamical system in the ( x, y)-plane. Assume that we specify the x
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