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.
