U.S.-Germany Cooperative Research on Attractive Invariant Manifolds in High-Dimensional Symmetric Systems

该奖项支持科罗拉多州立大学数学系的Michael Kirby和其他几位教授和研究生与德国图宾根大学理论物理研究所的Werner Guettinger教授等人在应用数学方面进行合作。他们合作的主题是研究多自由度非线性动力系统的复杂时空行为。他们的目标是将高维演化系统简化为低维动力系统，并用非线性动力学的定性几何概念来分析这些系统。要做到这一点，他们必须发展合适的方法来数学逼近不变流形。这项研究得益于德国小组在发展低维动力系统理论并将其应用于物理问题方面的开创性工作。联合研究还需要美国小组在降维和对称利用方面的互补专业知识。特别是，他们开发了基于神经网络建模算法的动态系统模式分析方法，从数学和物理的基本原理开始，这些合作科学家正在研究有组织结构和运动模式的发展，当外力超过临界值时，激活会被触发。他们试图证明对称或连贯的时空结构是如何产生的，以及为什么这种结构在自然界中更受欢迎。这些问题的答案也与更大的技术发展问题有关。

This award supports Michael Kirby, and several other professors and graduate students of the Department of Mathematics of Colorado State University, to collaborate in applied mathematics with Professor Werner Guettinger and others of the Institute for Theoretical Physics of the University of Tuebingen, Germany. The subject of their collaboration is the study of the complex spatio-temporal behavior of nonlinear dynamical systems with many degrees of freedom. Their goal is to reduce high dimensional evolution systems to low dimensional dynamical systems and to analyze those with the qualitative geometrical concepts of nonlinear dynamics. To do this they must develop suitable methods for mathematical approximation of invariant manifolds. The research benefits from the German group's pioneering work in developing low-dimensional dynamical systems theory and applying it to physical problems. The joint research also requires the complementary expertise of the US group in dimensionality reduction and symmetry exploitation. In particular, they have developed pattern analysis methods for dynamical systems based on neural network modeling algorithms Starting from fundamental principles of mathematics and physics, these collaborating scientists are investigating the development of organized structures and moving patterns whose activation is triggered when external forces exceed critical values. They seek to demonstrate how symmetrical or coherent spatio-temporal structures may arise, and why such structures are preferred in nature. The answers to such questions also have bearing on larger issues of technological development.