On Defining Chaos in the Absence of Time

  • Richard C. Churchill
Part of the NATO ASI Series book series (NSSB, volume 332)

Abstract

One of the standard definitions of a chaotic dynamical system on a metric space involves three conditions, one being sensitive dependence on the choice of initial values. Using the recent discovery that the sensitivity hypothesis is a logical consequence of the other two conditions we formulate a time-and-metric independent concept of chaos for foliations which implies the usual definition when the leaves are the orbits of a flow on a manifold. Simple examples are presented to make the point that any reference to “chaos” when either or both of the other standard conditions has not been verified could be quite misleading. In particular, for any integer n > 1 we give an example of a completely integrable n-degree of freedom Hamiltonian system, with compact energy surfaces, having the property that the induced flows on almost all energy surfaces admit sensitive dependence on initial conditions.

Keywords

Vector Field Periodic Orbit Sensitive Dependence Rest Point Chaotic Dynamical System 
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References

  1. [1]
    J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, “On Devaney’s definition of Chaos”, Am. Math. Monthly 99 (1991), pp. 332–334.MathSciNetCrossRefGoogle Scholar
  2. [2]
    R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd-ed., Addison-Wesley, Redwood City, CA., 1989.MATHGoogle Scholar
  3. [3]
    V.V. Nemytskii and V.V. Stepanov, Qualitative Theory of Ordinary Differential Equations, Princeton Univ. Press, Princeton, 1960.Google Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Richard C. Churchill
    • 1
  1. 1.Dept. of Mathematics, Hunter College and Graduate SchoolCUNYNew YorkUSA

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