Mathematical Sciences: A Combinatorial Method for Large Vortex Systems and Systematic Time-Reversible Dynamics in Particle Sedimentation

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Two separate methods for the study of dynamical properties of fluid-flows will develop in the context of specific problems. In part I, the quasi-periodic and chaotic motions of vortex systems with a large number of particles and unrestricted vorticity will be considered. These vortex systems include the point-vortex model, and desingularized versions such as the vortex blob model. A combinatorial approach that for the first time will enable the study of dynamical problems that are relevant vortex methods and particle methods in general to be used. This approach will overcome the severe restriction on the size of perturbations in Kolmogorov-Arnold-Moser theory, which has hitherto limited its applications to problems with few degrees of freedom. In part II, periodic and chaotic motions of stokes spheres in sedimentation theory will be studied. Near-equilibrium dynamics of identical sedimenting particles will emphasized. Equivariant version of the Melnikov theory will be attempted.