U.S.-Australia Cooperative Research on Symmetric Chaos

该奖项支持休斯顿大学园区分校马丁·戈卢比茨基博士和伊恩·墨尔本博士三次在三年多的时间里访问澳大利亚悉尼大学，与M.J.菲尔德博士在对称分叉的理论和应用方面的几个问题上合作。他们将研究对称映射的迭代(对称混沌)及其在常微分方程组和偏微分方程组中的表现，以及异宿循环的稳定性(自然出现在对称系统中)。尽管许多人研究了对称性对微分方程组的动力学和分叉的影响，很少有人注意到具有对称性的离散动力系统(在这种系统中，由于共轭混沌吸引子的碰撞和漂移而导致的对称性增加分叉的新现象沿着集团轨道运行似乎相当频繁)。研究人员将研究对称性增加的分叉与某些流体动力系统中的相干结构的存在有关的可能性，在某些流体动力系统中观察到对称性(平均)。例如，在Couette-Taylor实验(研究两个独立旋转的同心圆柱体之间的流体流动)中观察到了湍流Taylor涡。这种流体状态相当戏剧性地显示出平均而言具有独特模式的湍流运动共存，这很可能与混沌共存有关具有对称吸引子平均对称性的动力学。异宿周期与间歇和爆发现象的存在有关，因此提供了有趣的动力学。除了个别情况外，关于这类不变集的渐近稳定性的最优结果很少。研究人员建议调查到目前为止使用的技术是否可以统一起来，以提供更连贯的理论。

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.