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On Devaney’s definition of chaos. (English) Zbl 0758.58019
Although there has been no universally accepted mathematical definition of chaos, R. L. Devaney isolated three components as being its essential features: according to his definition [{\it R. L. Devaney}, An introduction to chaotic dynamical systems, 2nd ed. (1989; Zbl 0695.58002)], a continuous map $f: X\to X$, where $X$ is a metric space, is said to be chaotic on $X$ if 1) $f$ is transitive, 2) the periodic points of $f$ are dense in $X$, 3) $f$ has sensitive dependence on initial conditions. The aim of the paper is to prove the following result: if $f: X\to X$ is transitive and has dense periodic points then $f$ has sensitive dependence on initial conditions.

37D45Strange attractors, chaotic dynamics
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