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What is a chaotic attractor? (English) Zbl 1175.37039

The author discusses the notion of chaos and chaotic attractors. After T.-Y. Li and J. A. Yorke [Am. Math. Mon. 82, 985–992 (1975; Zbl 0351.92021)] first used the word chaos in deterministic systems, R. Devaney [An introduction to chaotic dynamical systems. Redwood City, CA etc.: Addison-Wesley Publishing Company (1989; Zbl 0695.58002)] gave the first mathematical definition for a chaotic map. Since then, several different definitions of chaos have been proposed, see [M. Martelli, M. Dang, and T. Seph, Math. Mag. 71, No. 2, 112–122 (1998; Zbl 1008.37014)] for an overview.
The author gives yet another definition (see the following Definition 3.3) of a chaotic diffeomorphism on an invariant set, and of a chaotic attractor, which is similar to the definition of M. Martelli [Introduction to discrete dynamical systems and chaos. New York, NY: Wiley (1999; Zbl 1127.37300)] and M. Martelli, M. Dang, and T. Seph [loc. cit.].
Definition 3.3. A diffeomorphism F is said to be chaotic on an invariant set A provided that:
(a) F has sensitive dependence on initial conditions when restricted to A.
(b) F is topologically transitive.
A set A is called a chaotic attractor for F if A is an attractor and F is chaotic on A.
The author gives several examples to illustrate his definition, in particular an example of a chaotic attractor in his sense which has zero entropy and is not an attractor in the sense of K. T. Alligood, T. D. Sauer, and J. A. Yorke [Chaos. An introduction to dynamical systems. New York, NY: Springer (1996; Zbl 0867.58043)].

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34C28 Complex behavior and chaotic systems of ordinary differential equations
34D45 Attractors of solutions to ordinary differential equations

Keywords:

chaos; attractor
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