Mathematical Sciences: Dynamics and Symmetry

小行星9403624 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，对称性是通过假定规则的几何形状或通过复杂系统中部件的可互换性引入的。 在实验和计算机模拟中，规则图案的存在通常是由于对称性。 直到最近才表明，对称性也可以强迫动力学中的不对称性和解或实验的时间平均模式的存在。 本提案的目的是调查这些现象和相关现象。特别是，我们将研究如何在动力系统中的对称性和混沌动力学interrupting，以及对称性如何影响耦合细胞系统中的解决方案的类型。 在这个项目中，我们将研究对称映射的迭代（对称混沌），每个子系统或细胞都有自己的内部对称性的常微分方程耦合系统的调查，异宿周期的稳定性（自然发生在微分方程的对称系统），以及强迫对称性破缺的影响（其中考虑对称方程对对称性较低的方程的扰动）。 在偏微分方程系统和实验（特别是泰勒-库埃特系统和法拉第表面波实验）中，平均模式的应用将被认为是在模拟多孔介质中对流的实验中存在的非线性（异宿周期）。 此外，一些具有对称性的微分方程的动力学将通过直接计算机模拟进行研究，并将结果与现有理论进行比较。

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.