On Defining Chaos in the Absence of Time

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.