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Deterministic dynamical systems and chaos. (English) Zbl 0893.58041

The dynamical systems which we consider here are given by a differentiable map \(f:X\to X\), \(X\) a finite-dimensional manifold. Sequences of the form \((x,f(x), f^2(x), \dots)\) are called evolutions. This means that we are thinking of \(f\) as the law of evolution of some system with discrete time (e.g., \(x,f(x), \dots\) might describe the successive sizes of generations of a population according to some model). We call these dynamical systems deterministic since the present determines the future.
For nonlinear \(f\) new types of evolutions become possible – these evolutions, which are bounded and which cannot be described in terms of a finite number of frequencies, are often called chaotic. Also, whenever all or many evolutions are chaotic, it is no longer true (at least in the known examples) that evolutions with nearby initial states stay for ever nearby – one speaks of sensitive dependence on initial conditions. This last phenomenon is related with the fact that evolutions which are bounded and chaotic are hard to predict.
In the following sections we describe two main examples of nonlinear systems with chaotic evolutions. Then we explain why these examples appear as subsystems in many (most?) nonlinear systems with chaotic evolutions, and how they are representatives of different classes of dynamical systems generating chaotic evolutions.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37F99 Dynamical systems over complex numbers
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