THE NATURAL MEASURE OF NONATTRACTING CHAOTIC SETS AND ITS REPRESENTATION BY UNSTABLE PERIODIC ORBITS

Abstract

The natural measure of a chaotic set in a phase-space region can be related to the dynamical properties of all unstable periodic orbits embedded in the chaotic set contained in that region. This result has been shown to be valid for hyperbolic chaotic invariant sets. The aim of this paper is to examine whether this result applies to nonhyperbolic, nonattracting chaotic saddles which lead to transient chaos in physical systems. In particular, we examine, quantitatively, the closeness of the natural measure obtained from a long trajectory on the chaotic saddle to that evaluated from unstable periodic orbits embedded in the set. We also analyze the difference between the long-time average values of physical quantities evaluated with respect to a dense trajectory and those computed from unstable periodic orbits. Results with both the Hénon map and the Ikeda–Hammel–Jones–Moloney map for which periodic orbits can be enumerated lend credence to the conjecture that the unstable periodic-orbit theory of the natural measure is applicable to nonhyperbolic chaotic saddles.