Mathematical Sciences: A study of chaos: theoretical and numerical investigations of transversal homoclinic points

在具有横截同宿点的系统中，混沌行为是最重要的情形之一。此外，这几乎是唯一可以严格证明混沌动力学存在的例子。在这个项目中，Palmer教授和Kocak教授将：1)扩展方程类，严格证明横截同宿点的存在性：2)发展特定方程中横截同宿点的数值方法；3)分析动力系统数值计算轨道的精度，并探讨双曲性在其中的作用。数学的发展将提供对物理和生物系统行为的洞察，在这些系统中，看似随机的动力学实际上受到确定性自然定律的支配。

One of the most important situations where chaotic behavior occurs is in systems with transversal homoclinic points. Moreover, this is almost the only instance for which the presence of chaotic dynamics can be rigorously established. In this project Professor Palmer and Kocak will 1) Enlarge the class of equations where the existence of transversal homoclinic points can be proved rigorously; 2) Develop numerical methods for the detection of transversal homoclinic points in specific equations; 3) Analyze the accuracy of numerically computed orbits of dynamical systems, and investigate the role of hyperbolicity therein. The mathematical development will provide insight into the behavior of physical and biological systems in which seemingly random dynamics are actually governed by deterministic natural laws.