Control and tracking of chaos in Hamiltonian systems

Control and tracking of chaotic Hamiltonian systems are addressed. We study the possibility of stabilization of chaotic behaviour of Hamiltonian systems with two degrees of freedom in general, and the classical diamagnetic Kepler problem, in particular. In order to stabilize the trajectories of a system it is necessary to have information about the position of its unstable periodic orbits. Two methods have been developed to detect the unstable periodic orbits in chaotic systems. We have applied these methods to a number of Hamiltonian systems successfully. A control algorithm based on the OGY method is applied to Hamiltonian systems. We have achieved for the first time, control of a number of classical unstable periodic orbits for the Hénon-Heiles system and the classical diamagnetic Kepler problem. The displacement of the position of an unstable periodic orbit as a system parameter is varied, is a major obstacle to maintain a longtime control. A new algorithm has been developed for tracking the position of unstable periodic orbits when the system parameter changes. We have shown the possibility of retaining control despite parameter variation.