Geometry and Computation of Dynamics for Conservative Systems

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Proposal #0202032PI: J.D. MeissInstitution: University of Colorado BoulderTitle: Geometry and Computation of Dynamics for Conservative SystemsABSTRACTThe principal investigator proposes to study the geometry of low-dimensional dynamical systems, especially symplectic and volume-preserving maps, using both computational and analytical techniques. While much is known about the two-dimensional case, there are still many questions about the onset and development of chaos for three- and higher-dimensional systems. While most oscillators are anharmonic (have twist), twistless bifurcations occur in one-parameter families of these systems. In the proposal, the geometry of twistless bifurcations will be studied leading to an understanding of fold and cusp bifurcations in the twist. The resulting geometry of the reconnection of resonances and exotic twistless tori will be studied numerically. These should play a role in limiting the stability domains for many dynamical systems. From the other side, the destruction of chaos can be profitably studied using a limit of extreme chaos, the anti-integrable (AI) limit as a starting point. In this proposal, the PI will use the AI limit to study coupled systems of maps and chaotic boundaries. Near this limit, structures such as exotic versions of the Smale horseshoe, and other heteroclinic tangles should occur. The onset of chaos in conservative systems is signaled by the destruction of tori. These have been studied by a rescaling analysis called the renormalization transformation. The structure of this transformation for four and higher dimensional systems is only beginning to be understood. The PI proposes that recent approximate versions of this transformation will give effective numerical strategies for finding the destruction and analyzing the topology of the resulting objects.Developing an understanding of the dynamics of conservative systems is important to applications including the design of particle accelerators, obtaining rates for simple chemical reactions, calculating confinement times for charged particles in plasma fusion devices, understanding the spectra of highly excited atomic systems, and designing efficient spacecraft trajectories in an era of lower budgets. Dynamics is such systems is often chaotic, which implies that prediction of specific trajectories is difficult; however, chaos can be profitably utilized to improve efficiency, for example of spacecraft trajectories, by judiciously applying small course corrections. Chaos can also dramatically affect the lifetimes of particles in confinement devices and the rates of chemical reactions. The PI proposes to develop geometrical and computational techniques that can be used to address these questions. In addition extending our understanding of chaos to higher dimensional cases will help populate the zoo of chaotic objects in multidimensional systems.