Stabilization of limit cycles in the Lorenz attractor through the orbit closure method

Abstract

This manuscript deals with the creation and stabilization of limit cycles in the Lorenz attractor with the help of orbit closure technique. 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. The orbit closure technique is a novel method of controlling chaos wherein a finite time control effort is applied to the system states, once per period, to bring back the chaotic trajectory to a desired periodic orbit so as to stabilize it. In this manuscript, the orbit closure technique is applied to create and stabilize limit cycles of different time periods in the Lorenz attractor. The control effort required for closure of the orbits is provided through feedback linearization. As a pilot study, the efficacy of the proposed technique is demonstrated through numerical simulations in this manuscript. Further, the influence of the control gain on the shape and time period of the stabilized orbit is studied and reported.