Mathematical Sciences: Dynamics and Symmetry

9403624戈尔比茨基在这个项目中，我们研究对称动力系统的理论和应用。在许多应用中，对称性要么是通过规则几何的假设引入的，要么是通过复杂系统中部件的互换性引入的。众所周知，在实验和计算机模拟中，规则图案的存在往往是由于对称性。直到最近才表明，动力学中间歇性的存在和解或实验的时间平均中的模式也可以被对称性所强迫。这项建议的目的是调查这些现象和相关现象。特别是，我们将研究对称性和混沌动力学如何在动力系统中交织在一起，以及对称性如何影响在耦合细胞系统中找到的解的类型。在这个项目中，我们将研究对称映射的迭代(对称混沌)，研究耦合常微分方程组，其中每个子系统或单元都有自己的内部对称性，异宿环的稳定性(这自然发生在对称微分方程组中)，以及强迫对称破缺的影响(其中考虑对称方程到较不对称的方程的扰动)。我们将考虑在偏微分方程组和实验(特别是在Taylor-Couette系统和Faraday表面波实验中)中的平均模式的应用，以及在模拟多孔介质中的对流的实验中是否存在间歇(异宿循环)。此外，还将通过直接的计算机模拟来研究一些具有对称性的微分方程的动力学，并将结果与现有的理论进行比较。

9403624 Golubitsky In this project we study both the theory and application of symmetric dynamical systems. In many applications, symmetry is introduced either by the supposition of a regular geometry or by the interchangability of parts in a complex system. It has been well established that the existence of regular patterns in experiments and in computer simulations are often due to symmetry. Only recently has it been shown that the existence of intermittency in the dynamics and patterns in the time-average of solutions or experiments can also be forced by symmetry. It is the purpose of this proposal to investigate these and related phenomena. In particular we will study how symmetry and chaotic dynamics intertwine in dynamical systems and how symmetry affects the types of solutions found in systems of coupled cells. In this project we will study iteration of symmetric maps (symmetric chaos), the investigation of coupled systems of ODEs where each subsystem or cell has its own internal symmetry, the stability of heteroclinic cycles (which occur naturally in symmetric systems of differential equations), and the effect of forced symmetry breaking (where perturbations of a symmetric equation to one with less symmetry are considered). Applications to patterns on average in systems of PDEs and experiments (particularly in the Taylor-Couette system and the Faraday surface wave experiment) will be considered as will the existence of intermittency (heteroclinic cycles) in an experiment modeling convection in a porous media. In addition, the dynamics of a number of differential equations with symmetry will be studied by direct computer simulation and the results compared with current theory.