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A characterization of chaos

Published online by Cambridge University Press:  17 April 2009

K. Janková  and
J. Smítal
K. Janková
Affiliation:
Department of Mathematics, Komensky University, 842 15 Bratislava, Czechoslovakia.
J. Smítal
Affiliation:
Department of Mathematics, Komensky University, 842 15 Bratislava, Czechoslovakia.
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Abstract

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Consider the continuous mappings f from a compact real interval to itself. We show that when f has a positive topological entropy (or equivalently, when f has a cycle of order ≠ 2n, n = 0, 1, 2, …) then f has a more complex behaviour than chaoticity in the sense of Li and Yorke: something like strong or uniform chaoticity, distinguishable on a certain level ɛ > 0. Recent results of the second author then imply that any continuous map has exactly one of the following properties: It is either strongly chaotic or every trajectory is approximable by cycles. Also some other conditions characterizing chaos are given.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 34 , Issue 2 , October 1986 , pp. 283 - 292
Copyright
Copyright © Australian Mathematical Society 1986

References

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