U.S.-Mexico Collaborative Research: Dynamics of Extended Systems and Coupled Map Lattices

这个美国墨西哥奖将支持Mark Levi、Howard Weiss和Yakov Pesin博士与圣路易斯·德·波托西大学数学系的瓦伦丁·阿弗拉莫维奇合作进行研究。研究人员计划研究：1)来自物理学和生物学的偏微分方程对应的耦合映射格子，特别是Fitzhugh-Nagumo方程(这是神经生物学中非常感兴趣的)。合作者打算描述其局部映射的遍历性质，并构造该映射的吸引子的SRB度量。2)通过行波将耦合映象格子转化为偏微分方程组，为发展型偏微分方程组的数值模拟奠定了基础。3)耦合振子链的动力学，特别是与Sine-Gordon方程有关的动力学。耦合映象格子是一个离散的时间动力学系统，它的相空间具有特定的形式，并且整个系统具有平移对称性。耦合映象格子最近作为时空混沌和相干结构的模型得到了广泛的应用。目前，人们对使用耦合映射格子来模拟湍流、神经细胞、统计物理中的相变和晶体非常感兴趣。它们也自然地产生于演化偏微分方程式的离散版本。

0104675LeviThis U.S. Mexico award will support Drs. Mark Levi, Howard Weiss, and Yakov Pesin in a research collaboration with Valentine Afraimovich of the Mathematics Department of the Unversidad Autonoma de San Luis de Potosi. The investigators plan to study: 1) Coupled map lattices corresponding to partial differential equations from physics and biology, in particular, the FitzHugh-Nagumo equation (which is of great interest in neurobiology). The collaborators intend to describe the ergodic properties of its local map and construct SRB measures for the attractor of this map. 2) The transition from coupled map lattices to partial differential equations via traveling waves, which will build a foundation for numerical modeling of some partial differential equations of evolution type. 3) The Dynamics of chains of coupled oscillators, in particular, those associated with the Sine-Gordon equation.A coupled map lattice is a discrete time dynamical system whose phase space is of a particular form, and for which the overall system exhibits translational symmetry.Coupled map lattices have recently gained wide popularity as models of spatio-temporal chaos and coherent structures. There is currently great interest in using coupled map lattices to model turbulence, nerve cells, phase transitions in statistical physics, and crystals. They also arise naturally from the discrete version of evolution partial differential equations.