Mathematical Sciences: Transport for Symplectic Mapping

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This project will study transport phenomena in symplectic mappings of four or more dimensions. The goal is to partition phase space into regions bounded by partial barriers through which flux occurs. A candidate for such a region is a resonance, defined as a volume in the neighborhood of an elliptic point; its boundary will be obtained from limits of librational periodic orbits. Resonance volumes will be computed and we will determine whether resonances partition phase space. The correlation function for orbits in the neighborhood of a resonance boundary well be studied to determine whether the series for the diffusion tensor converges. Transport will be defined in terms of the flux across resonance boundaries, including branching ratios for transitions from one resonance to a neighboring one, as well as drift along a commensurability channel. To compute the latter a restricted symplectic map will be introduced and studied. Numerical techniques will include frequency filtering and finding periodic orbits.