Regular-to-irregular transition in conservative Hamiltonian systems: Critical energies and local entropies

Abstract

The energy dependence of trajectories in the neighborhood of hyperbolic points is studied for a variety of two-degree-of-freedom conservative classical Hamiltonian systems displaying a "stochastic transition." In all cases studied the energy onset of substantial irregularity (defining a critical energy ${E}_{c}$) is shown to occur when local entropies, defined in terms of generalized characteristic multipliers, equal unity. Similar results are obtained for the standard map. The results suggest a unifying quantity for describing the onset of substantial irregularity in two-degree-of-freedom systems.