U.S.-Australia Cooperative Research on Symmetric Chaos

点击翻译按钮获取中文摘要

This award supports three visits by Dr. Martin Golubitsky and three by Dr. Ian Melbourne, of the University of Houston, University Park, over three years, to the University of Sydney, Australia, to work with Dr. M.J. Field on several problems in both the theory and application of bifurcations in the presence of symmetry. They will study the iteration of symmetric maps (symmetric chaos) and its manifestations in systems of ordinary and partial differential equations, and the stability of heteroclinic cycles (which occur naturally in symmetric systems). Although many have studied the effect of symmetry on both the dynamics and bifurcation of systems of differential equations, few have paid attention to discrete dynamical systems with symmetry (where the new phenomenon of symmetry-increasing bifurcations forced by the collision of conjugate chaotic attractors and drifting along group orbits seems to occur rather frequently). The researchers will investigate the possibility that symmetry increasing bifurcations are related to the existence of coherent structures in certain fluid dynamical systems where symmetry (on average) is observed. For example, turbulent Taylor vortices are observed in the Couette- Taylor experiment (in which fluid flow between two independently rotating concentric cylinders is studied). This fluid state demonstrates rather dramatically the coexistence of turbulent motion with distinctive patterns on average which may well be related to the coexistence of chaotic dynamics with the symmetry on average of symmetric attractors. Heteroclinic cycles are associated with the existence of intermittency and bursting phenomena, thus providing interesting dynamics. Except in isolated cases, there are few optimal results concerning the asymptotic stability of such invariant sets. The researchers propose to investigate the possibility that the techniques used so far can be unified to give a more coherent theory.