Symmetry, Bifurcations and Dynamics

NSF奖摘要-DMS-0071735数学科学：对称性、分叉和动力学这个项目将研究等变动力系统的理论和应用。特别是，利用微分方程和模型的对称性，我们将研究在空间和时间上都发生的模式。我们将探索两足动物和四足动物的运动中心模式生成器，突发类型的分类，螺旋波的超弯曲，以及磁性发电机。我们还将研究欧几里得对称性和混沌动力学的结合，这将导致随机布朗运动的确定性机制，以及空间扩展系统的Ginzburg-Landau理论。图案出现在空间和时间的物理、化学和生物系统中，并且是醒目的和可重复的。通常这些模式是由对称性产生的，我们的研究将通过进一步发展对称动力系统的理论和应用来研究模式的形成。将被研究的应用范围从动物步态中的肌肉节奏和磁性发电机中的间歇性，到产生突发信号和螺旋波运动的复杂方式。

NSF Award Abstract - DMS-0071735Mathematical Sciences: Symmetry, Bifurcations, and DynamicsAbstract0071735 Golubitsky This project will investigate the theory and applications of equivariant dynamical systems. In particular, using the symmetries of differential equations and models, we will study patterns that occur both in space and in time. We will explore locomotor central pattern generators for bipeds and quadrupeds, a classification of bursting types, hypermeander of spiral waves, and magnetic dynamos. We will also study a combination of Euclidean symmetry and chaotic dynamics that leads to a deterministic mechanism for stochastic Brownian motion, and Ginzburg-Landau theory for spatially-extended systems.Patterns appear in physical, chemical, and biological systems in both space and time and are striking and reproducible. Often these patterns result from symmetries, and our research will study pattern formation by further developing the theory and application of symmetric dynamical systems. Applications that will be investigated range from the study of muscle rhythms in animal gaits and intermittency in magnetic dynamos to the complicated ways in which bursting signals are produced and spiral waves move.