CHAOTIC TRANSPORT IN THE HOMOCLINIC AND HETEROCLINIC TANGLE REGIONS OF QUASI-PERIODICALLY FORCED 2-DIMENSIONAL DYNAMIC-SYSTEMS

We generalize notions of transport in phase space associated with the classical Poincare map reduction of a periodically forced two-dimensional system to apply to a sequence of non-autonomous maps derived from a quasiperiodically forced two-dimensional system. We obtain a global picture of the dynamics in homoclinic and heteroclinic tangles using a sequence of time-dependent two-dimensional lobe structures derived from the invariant global stable and unstable manifolds of one or more normally hyperbolic invariant sets in a Poincare section of an associated autonomous system phase space. The invariant manifold geometry is studied via a generalized Melnikov function (Wiggins). Transport in phase space is specified in terms of two-dimensional lobes mapping from one to another within the sequence of lobe structures, which provides the framework for studying several features of the dynamics associated with chaotic tangles. Instantaneous and average flux with respect to a sequence of cores are quantified analytically in the near-integrable case through the use of the generalized Melnikov function, and the computational prescription for determining flux in systems that are not near-integrable is provided. Other basic transport issues (both in the near-integrable and general case) are studied by a consideration of the geometry of intersections of lobes with each other, with the core, and with level sets of the unperturbed Hamiltonian. We compare transport rates in the single and multiple frequency cases. Though the comparisons depend on choices of normalization, for some reasonable normalizations the average flux of a class of multiple frequency systems is found to be maximal in a certain single frequency limit. The variation of lobe areas in multiple frequency systems, however, gives one the freedom to enhance or diminish aspects of transport over a finite time scale for a fixed average flux. Numerical simulations of lobe structures are presented, using a double phase slice method (Beigie et al), which provides the basis for exact computation of lobe areas and other transport quantities. The chaotic nature of the dynamics is understood in the framework of a travelling horseshoe map sequence. The generalized Melnikov function provides a tool to search through parameter space to determine when a quasiperiodically forced system is chaotic. The extension of the analysis to more general time dependences is explained.