Computation of Periodic Orbits and Verification of Chaos in Dynamical Systems

研究者和他的同事使用阴影技术为混沌动力系统设计了严格的计算方法。特别是，他们开发了确定周期轨道及其李雅普诺夫指数的算法。考虑到周期轨道的稳定流形和不稳定流形在确定全局动力学中的重要作用，提出了一种可靠计算周期轨道流形的方法。此外，他们还开发了一种仅从有限计算轨道建立混沌行为的方法和一种测量初始数据灵敏度的算法，这是混沌的标志。所有的计算算法都是在自治常微分方程的背景下发展起来的。尽管动力系统理论最近取得了重大进展，但在应用中出现的特定方程的分析仍然必须在很大程度上依赖于数值计算。因此，设计可验证其可靠性的计算技术是很重要的。在混沌系统的数值研究中，严格性的必要性尤其突出，这自然会放大很小的计算误差。混沌系统在令人眼花缭乱的复杂性中，拥有许多不稳定的周期轨道。该项目的主要目标是开发定位这种周期轨道及其相关结构的计算方法。混沌系统的周期轨道虽然不稳定，但在航天器的飞行路径设计等方面具有重要的实际意义。研究人员正开始与喷气推进实验室的科学家合作，对合适的路径进行高效可靠的计算。

The investigator and his colleague use the technique of shadowing to devise rigorous computational methods for chaotic dynamical systems. In particular, they develop algorithms for the determination of periodic orbits and their Lyapunov exponents. Given the significant role the stable and unstable manifolds of periodic orbits play in determining the global dynamics, a method to compute these manifolds reliably is developed. Moreover, they develop a method of establishing chaotic behavior just from finite computed orbits and an algorithm for measuring sensitivity to initial data, which is the hallmark of chaos. All the computational algorithms are developed in the context of autonomous ordinary differential equations. Despite the recent major advances in dynamical systems theory, analysis of specific equations arising in applications must still largely rely on numerical computations. Therefore, it is important to devise computational techniques whose reliability can be verified. Indispensability of rigor is particularly acute in the numerical studies of chaotic systems, which naturally amplify even small computational errors. Chaotic systems possess, in the mist of bewildering complexity, many unstable periodic orbits. The main goal of the project is to develop computational methods for locating such periodic orbits and their associated structures. Periodic orbits of chaotic systems, although unstable, are of practical significance in, for example, designing flight paths for spacecraft. The investigators are beginning to cooperate with Jet Propulsion Laboratory scientists on efficient and reliable computations of suitable paths.