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Zbl 0758.58019
Banks, J.; Brooks, J.; Cairns, G.; Davis, G.; Stacey, P.
On Devaney's definition of chaos. (English)
[J] Am. Math. Mon. 99, No.4, 332-334 (1992). ISSN 0002-9890

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.
[I.Oprea (Bucureşti)]

MSC 2000:: *37D45 Strange attractors, chaotic dynamics

Keywords: dynamical system; chaos; dependence on initial conditions

Citations: Zbl 0695.58002

Cited in: Zbl 1062.37007 Zbl 0891.58027 Zbl 0866.58046

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