Mathematical Sciences: Transition to Chaos in Multidimensional Hamiltonian Systems

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9305847 Meiss The dynamics of four and higher dimensional symplectic mappings is of fundamental importance to understanding stability and chaos in conservative physical systems. In this proposal a combination of numerical and analytical techniques will be used. We propose to determine the domain of existence of invariant tori both by using recursive generation of the Fourier series for the tori, and by continuation of the Cantor sets from the anti-integrable limit. The goal is to develop methods for estimating practical stability boundaries and for investigating the transition to chaotic behavior. Computations will determine the robustness of the tori of various frequency vectors, leading to a generalization of the noble numbers that provide the most robust frequencies in two dimensions. A study of one dimensional, resonant tori will also be undertaken-these may be more persistent than two-tori, and form an important component of the barriers to transport. Transport in four dimensions. will be studied by numerical computation of exit time decompositions for cylinders of various homotopy types. Our goal is the development of a geometrical description of trapping regions and resonance zones and a characterization of the practical stability domain around an elliptic point. New techniques for control of transport will be developed for symplectic systems. All of the fundamental equations of physics are formulated as Hamiltonian dynamical systems. We propose to study the structure of the orbits of these systems with the motivation being to understand the problem of "transport." This is of primary importance in such areas as particle accelerator confinement, chemical reaction rates, fluid mixing, plasma confinement in magnetic fusion devices, asteroid and planetary ring stability, etc. The basic question is: how does a system evolve from one state (e.g. a confined beam in an accelerator), to another (e.g. beam hits the tunnel wall), and how long does this take. Typi cally trajectories must wend their way through exotic structures such as Cantor sets and self-similar fractals, some of which exhibit a remarkable "stickiness", in order to move through the phase space. The construction and visualization of these structures requires careful computer study guided by mathematical insight. A major problem is that the systems of interest correspond to four and higher dimensional spaces--our ordinary three-dimensional intuition fails. In various applications transport is either to be encouraged (speeding up reaction rates) or discouraged (confining particles); we will investigate techniques for accomplishing both tasks. ***