Chaos and Bifurcations in Volume-Preserving Dynamics

University Of Colorado At Boulder, Boulder CO

Investigators

Abstract

Abstract Regular, quasiperiodic motion is ubiquitous in dynamical systems with sufficient symmetry. A prominent example occurs in the Hamiltonian or symplectic case, where these "invariant tori" persist---even for nearly-integrable motion, as is explained by "KAM theory." The destruction of tori in the two-dimensional case is explained by Aubry-Mather theory and renormalization results. However, a concomitant understanding of the destruction of tori upon perturbation in higher dimensions has proved elusive. In this proposal, the implications of integrability, due to symmetries and invariants, of volume-preserving dynamics will be investigated. The loss of integrability under perturbation will be studied by a combination of analytical (Aubry's anti-integrable limit, Fourier series) and numerical (invariant manifold and continuation) techniques. Tori are both created and destroyed by bifurcations, and a study of the normal forms for codimension-one and two bifurcations of fixed points will lead to classification possible phenomena. Transport will be investigated numerically with the goal of developing analytical measures of flux and transport distributions. In a second project, the PI will investigate bifurcations in nonsmooth systems appropriate to the modeling of chemical reactions, the systematic simplification of these systems by center manifold reduction, as well as the study of transport caused by weak coupling of chaotic motion to regular motion. Conservative dynamical models are used in designing particle accelerators, obtaining rates for simple chemical reactions, calculating confinement times in plasma fusion devices, understanding the spectra of highly excited atomic systems, and designing efficient spacecraft trajectories. Dynamics in such systems is often chaotic and prediction of individual trajectories is difficult; nevertheless, chaos can be profitably utilized, for example, to improve efficiency of spacecraft trajectories, by judiciously applying small course corrections, or to enhance the lifetimes of particles in confinement devices and the rates of chemical reactions. Volume-preserving dynamics models the flow of incompressible fluids and magnetic fields and a quantitative understanding of chaos in these systems is crucial for the development of efficient mixing in microscale bioreactors as well as of predictive planetary scale weather models. Most of our current theoretical understanding is limited to the two-dimensional case that is appropriate for flows in rapidly rotating or thin layers of fluid. While this has been useful in the understanding of such phenomena as the trapping of nutrients in gulf stream rings, the formation of the ozone hole and the creation of vortex-induced mixing in sinuous tubes, even in these systems, three-dimensional, chaos-induced transport needs to be understood. The PI seeks to develop analytical and computational methods for the study of regular and chaotic volume-preserving motion both to contribute broadly to our fundamental understanding of the richness of the behavior of low-dimensional deterministic evolution, and, to relate it to mixing and transport.