Symmetry, Bifurcations and Dynamics

数学科学：对称、分岔和动力学[Abstract] 0071735 Golubitsky该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。我们将探讨两足动物和四足动物的运动中枢模式发生器、爆发类型的分类、螺旋波的超曲流和磁发电机。我们还将研究欧几里得对称和混沌动力学的结合，这导致了随机布朗运动的确定性机制，以及空间扩展系统的金兹堡-朗道理论。模式出现在空间和时间的物理、化学和生物系统中，是引人注目的和可复制的。这些图案通常是由对称性产生的，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图案的形成。将被研究的应用范围从动物步态的肌肉节律和磁发电机的间歇性研究，到爆发信号产生和螺旋波移动的复杂方式。

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.