Creation and stabilization of limit cycles in chaotic attractors through closure of orbits

This manuscript presents a novel method of creating and stabilizing limit cycles in a chaotic attractor. Chaos control techniques such as the OGY and OPF techniques apply small perturbations to a system parameter to stabilize an unstable periodic orbit present in a chaotic attractor. But it may happen that the system parameters may not be available for control. For example, in structural systems, it is difficult to alter their geometry and material properties in real-time, so as to suppress chaos. In such cases, use of state feedback control will provide ease of implementation. But impulse-like small perturbations, prescribed by the OGY technique, applied as a feedback to the state equations cannot drive the system from its current state to the desired state. A proper control law must be devised to drive the system from one state to another. Hence, we propose that a finite time control effort can be applied to the system states, once per period, to bring back the chaotic trajectory to the desired orbit so as to stabilize it. The implementation of the proposed technique is demonstrated with the help of a bistable Duffing oscillator in this manuscript. It is shown through numerical simulations that the proposed technique is able to stabilize periodic orbits passing through a prescribed set of points lying in the chaotic attractor of a Duffing oscillator.