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Transport in two-dimensional maps. (English) Zbl 0755.58070
Summary: We study transport in the two-dimensional phase space of \(C^ r\) diffeomorphisms \((r\geq 1)\) of two manifolds between regions of the phase space bounded by pieces of the stable and unstable manifolds of hyperplane fixed points. The mechanism for the transport is associated with the dynamics of homoclinic and heteroclinic tangles, and the study of this dynamics leads to a general formulation for the transport rates in terms of distributions of small regions in phase space (“lobes”). It is shown how the method applies to three geometrical configurations, one of which corresponds to the geometry associated with the Kelvin-Stuart Cat’s Eye flow undergoing a time-periodic perturbation. In this case the formulae imply, for example, that the evolution of only two lobes determines the mass transport from the upper to the lower half plane of the fluid flow. As opposed to previous studies this formulation takes into account the effect of re-entrainment of the lobes, i.e., the implications of the lobes leaving and re-entering the specified regions on the transport rates. The formulation is developed for both area- preserving and non-area preserving two-dimensional diffeomorphisms and does not require the map to be near-integrable. The techniques involved in applying this formulation are discussed including the possible use of the generating function for computing the distributions of the lobes in phase space, as well as the use of Poincaré maps, which enable one to study the transport in continuous time systems using the above formalism. In particular, we demonstrate how the right choice of the Poincaré section can reduce the labor of transport rate calculations.

MSC:
58Z05 Applications of global analysis to the sciences
82C70 Transport processes in time-dependent statistical mechanics
37N99 Applications of dynamical systems
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