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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 [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.

##### MSC:
 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
##### Keywords:
dynamical system; chaos; dependence on initial conditions
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