Dynamical correlations and chaos in classical Hamiltonian systems

Abstract

An improved local criterion is proposed for the onset of dynamical chaos in a classical mechanical Hamiltonian system. This criterion is based on the observation that it is only neighboring trajectories which diverge in the direction perpendicular to the flow in phase space that contribute to chaotic motion. The necessary condition for such divergence of neighboring trajectories is the existence of a region where the curvature of the potential energy surface perpendicular to the trajectory is negative. It is shown, for the Hénon–Heiles model system, that the lowest critical energy for the onset of chaos is the same as obtained from the Brumer–Duff–Toda–Corjan–Reinhardt criterion, but that the regions of prescribed negative curvature are only a small fraction of the accessible energy surface. Consequently, the BDTCR criterion for onset of chaotic motion is augmented by the condition that the trajectory cross the critical locus in a manner which permits penetration of one of the regions where the surface curvature is negative perpendicular to the trajectory. Given that the regions of local instability are only a fraction of the accessible energy surface, the relationship between local instability and global dynamical chaos is discussed, and a mapping of the trajectory into a macrostate occupation number representation proposed. That mapping is used to generate an upper bound for the Kolmogorov entropy of the system. The upper bound so obtained is only close to the Kolmogorov entropy when the sequential occupation of macrostates by a trajectory consists of successive uncorrelated transitions. For the Hénon–Heiles model it is shown that correlation in macrostate occupation along a trajectory is long lived, so that even when there is dynamical chaos on the long time scale there can be quasiperiodic motion on the short time scale. The relationship of the new results to the characterization of intramolecular energy transfer is briefly discussed.