Invariant manifolds and control of hyperbolic trajectories on infinite- or finite-time intervals

Abstract

We consider general non-autonomous systems on infinite- and finite-time intervals and describe some properties of hyperbolic trajectories and their stable and unstable manifolds. Our definitions of hyperbolic trajectories and their stable and unstable manifolds on finite-time intervals are different from one adopted in the previous references, but still possess a desirable property as the previous one. Furthermore, we present numerical methods based on these theoretical results to compute the stable and unstable manifolds, and propose a control method to stabilize unstable hyperbolic trajectories using geometrical structures near them like the Ott, Grebogi and Yorke (OGY) chaos control method. To demonstrate our methods, we give numerical computation results for two examples: a controlled pendulum on infinite- and finite-time intervals and a simple model for a spacecraft transferring from the Earth to the Moon.