Mathematical Sciences: Dynamics and Symmetry

在这个项目中，我们研究了对称动力系统的理论和应用。在许多应用中，对称性要么是由规则几何的假设引入的，要么是由复杂系统中零件的可互换性引入的。在实验和计算机模拟中，规则模式的存在通常是由于对称性，这一点已经得到了很好的证实。直到最近才有研究表明，动力学中的间断性和解或实验的时间平均模式的存在也可以被对称性所强迫。本提案的目的是调查这些和相关的现象。特别地，我们将研究对称性和混沌动力学如何在动力系统中交织，以及对称性如何影响耦合单元系统中发现的解的类型。在这个项目中，我们将研究对称映射（对称混沌）的迭代，对每个子系统或单元都有自己内部对称性的ode耦合系统的研究，异斜循环的稳定性（在微分方程的对称系统中自然发生），以及强制对称性破缺的影响（考虑对称方程对对称性较小的方程的扰动）。应用于偏微分方程系统和实验（特别是Taylor-Couette系统和法拉第表面波实验）的平均模式，以及在多孔介质中模拟对流的实验中间歇性（异斜周期）的存在将被考虑。此外，本文还将采用直接计算机模拟的方法研究一类具有对称性的微分方程的动力学问题，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.