Chaotic Hamiltonian Dynamics.

Chaotic behavior may be observed in deterministic Hamiltonian Systems with as few as three dimensions, i.e., X, P, and t. The amount of chaotic behavior depends on the relative influence of the integrable and non-integrable parts of the Hamiltonian. The Standard Map is such a system and the amount of chaotic behavior may be varied by adjusting a single parameter. The global phase space portrait is a complicated mixture of quiescent and chaotic regions. First a new calculational method, characterized by a Fractal Diagram, is presented. This allows the quantitative prediction of the boundaries between regular and chaotic regions in phase space. Where these barriers are located gives qualitative insight into diffusion in phase space. The method is illustrated with the Standard Map but may be applied to any Hamiltonian System. The second phenomenon is the Universal Behavior predicted to occur for all area preserving maps. As a parameter is varied causing the mapping to become more chaotic a pattern is observed in the location and stability of the fixed points of the maps. The fixed points undergo an infinite sequence of period doubling bifurcations in a finite range of the parameter. The relative locations of the fixed point bifurcation and the parameter intervals between bifurcations both asymptotically approach constants which are Universal in that the same constants keep appearing in different problems. Predictions of Universal Behavior have been based on the study of algebraic mappings. The problem we examine has a Hamiltonian given by H = p^2 over {2} - lambda over{2pi}sin(2pi x)sin(2pit). This Hamiltonian describes the motion of a compass needle in a sinusoidally varying magnetic field or, equally well, the one dimensional motion of a particle in a standing wave potential. By treating the magnitude(lambda ) of the time dependent potential as a parameter and by examining the trajectories of the system in a Poincare surface of section, the resulting differential equations may be reduced to an iterated mapping. The predicted Universal Behavior is observed in this problem for both oscillatory (trapped) and rotational (untrapped) motion. This particular problem also exhibits a sequence of restabilizations and destabilizations beyond the accumulation point of the period doubling bifurcation sequence. (Abstract shortened with permission of author.).