On Devaney’s definition of chaos. (English) Zbl 0758.58019

Banks, J.; Brooks, J.; Cairns, G.; Davis, G.; Stacey, P.

Am. Math. Mon. 99, No. 4, 332-334 (1992).

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.

Reviewer: I.Oprea (Bucureşti)

MSC:

37D45

Strange attractors, chaotic dynamics of systems with hyperbolic behavior

Keywords:

dynamical system; chaos; dependence on initial conditions

Citations:

Zbl 0695.58002