U.S.-Australia Cooperative Research on Symmetric Chaos

该奖项支持马丁博士的三次访问 Golubitsky和三个由伊恩墨尔本博士， 休斯顿大学，大学公园， 三年多来，悉尼大学， 澳大利亚，与M. J.菲尔德博士合作， 理论和实践中存在的几个问题 分支的应用 对称性 他们将研究 对称映射（对称混沌）及其 在普通和 偏微分方程， 异宿环的稳定性（ 自然地在对称系统中）。 虽然 许多人研究了对称性对 系统的动力学和分岔 在微分方程中，很少有人 注意离散动力系统， 对称性（其中， 强迫分叉 共轭混沌吸引子的碰撞， 似乎会发生沿着群轨道漂移 比较频繁）。 研究人员将 研究对称性 增加的分叉与 相干结构的存在 流体动力学系统，其中对称性（在 平均值）。 例如，湍流 在库埃特- 泰勒实验（流体在 两个独立旋转的同心 气缸进行了研究）。 这种流体状态 相当戏剧性地展示了 湍流运动与 平均而言， 与混沌共存有关 动力学的对称性， 对称吸引子 异宿环是 与不稳定性的存在有关 和爆裂现象，从而提供 有趣的动态 除了孤立的 情况下，很少有最佳结果 关于这样的渐近稳定性 不变集 研究人员建议， 调查一下 到目前为止使用的技术可以统一起来， 一个更连贯的理论。

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.