Critical point analysis of instabilities in Hamiltonian systems: Classical mechanics of stochastic intramolecular energy transfer

Abstract

A critical point analysis of the nonlinear dynamics of the difference between two neighboring classical trajectories is used to develop a simple criterion for the occurrence of stochastic instabilities in conservative classical Hamiltonian systems. The technique is based on the assumption that a necessary (but not sufficient) condition for existence of macroscopic stochastic regions in phase space follows from exponentiation of such neighboring trajectories, and that the critical (or stationary) points of the difference flow controls the global behavior. The method is successfully applied to the one-dimensional cubic and Morse oscillators, and to the two-dimensional (four degrees of freedom) Henon–Heiles, Barbanis, and anti-Henon–Heiles problems. The relationship to the earlier Brumer–Duff–Toda (BDT) method, which often makes incorrect predictions of instability, is established by noting that the BDT analysis, by linearizing the difference trajectory dynamics, ignores all but one of the flow controlling critical points.