Chaos control in the standard map aiming future applications in astrodynamics

Abstract

Non-linear dynamical systems model several natural phenomena and engineering problems. The solutions of some of these systems have chaotic behavior, that is, they present sensitive dependence on initial conditions. One such system, which is widely studied, is the Restricted Three-Body Problem. This model displays chaotic dynamics and has applications in Astrodynamics, such as, for example, low-energy Earth-Moon transfers and capture of asteroids. Sometimes, it is desirable to control the chaotic solutions in these systems by using a procedure known as chaos control, through which a small perturbation is applied to a chaotic orbit causing it to behave in a periodic manner. This project aims to study and implement the chaos control method known as OGY due to its creators E. Ott, C. Grebogi and J.A. Yorke. The method will be investigated in the Standard Map, a well-known mathematical model to which several dynamic systems can be locally reduced, in order to understand the role of hyperbolic eigendirections in the behavior of controlled solutions. In the future, it is expected to advance the study towards more general control schemes that can be applied to Poincaré maps of the Restricted Three-Body Problem aiming to stabilize the movement of a spacecraft around the equilibrium points of the Earth-Moon system.