Abstract :
Invariant manifold formulations of phase-space transport theory offer a well-known means of prediction for the chaotic dynamics associated with - and 2-degree-of-freedom systems. Whether such a means of prediction extends to systems with more than 2 degrees-of-freedom has thus far received little attention. We derive a multi-degree-of-freedom turnstile-based phase space transport theory for Hamiltonian flows with non-toroidal intersections of global stable and unstable manifolds of invariant normally hyperbolic sets of a Poincaré section on the energy surface. These non-toroidal manifold intersections are the typical case for flows with more than 2 degrees-of-freedom, and they disallow a partitioning of phase space via segments of stable and unstable manifolds alone. We show it is nevertheless possible for these intersection geometries to construct codimension-one partitions (dividing surfaces) of the Poincaré section that respect asymptotic behavior, and hence divide between regions of qualitatively different dynamics (bounded versus unbounded, rotation versus libration, and so on). These partitioning surfaces thus provide a geometrically exact criterion for capture into, and escape from, regions in the Poincaré section corresponding to qualitatively different dynamics. A turnstile-based phase space transport theory is then derived to describe transitions across the partitioning surfaces. We use Melnikov theory to study partition geometry, and to search through multi-dimensional phase space and parameter space to explore variation of O(ε) flux across the partition. Searches of this kind would be helpful in the control and optimization of any number of nonlinear physical processes associated with transitions across broken separatrices, such as molecular dissociation and atomic ionization. We observe significant variation in the effectiveness of separatrix crossing with differing phase and action values, and hence note the motivation to retain phase information, and the role of Arnold diffusion in sending points to regions of enhanced or diminished separatrix crossing, referred to as Arnold enhancement and Arnold suppression of transport across codimension-one partitions.
