Global effects of Pyragas time-delayed feedback control

Abstract

Pyragas time-delayed feedback is a control scheme designed to stabilize unstable periodic orbits, which occur naturally in many nonlinear dynamical systems. The control scheme targets a speci c unstable periodic orbit by adding a feedback term with a delay chosen as the period of the unstable periodic orbit. In this thesis we consider the global e ects of applying Pyragas control to a nonlinear dynamical system near a subcritical Hopf bifurcation. We start by considering the standard example of the Hopf normal form subject to Pyragas control, which is a delay di erential equation that models how a generic unstable periodic orbit is stabilized. We nd that the addition of feedback induces in nitely many Hopf bifurcation curves and possibly in nitely many stable periodic orbits in addition to the target periodic orbit. Therefore, the controlled system could follow one of these periodic orbits rather than the target periodic orbit. As such, we nd that to ensure successful implementation of the control scheme, one must consider the global dynamics of the system. Furthermore, we consider the e ect of a delay mismatch in the system, where the delay is set close to but not equal to the period of the target periodic orbit. We nd that the delay must be set as at least a linear approximation of the period of the target periodic orbit. To verify the predictiveness of the normal form analysis, we consider the global dynamics of the Lorenz system subject to Pyragas control. We nd that the addition of feedback induces further Hopf bifurcation curves and further stable periodic orbits, showing that the Hopf normal form with feedback is indeed predictive of the observed global dynamics and the e ect of a delay mismatch in the system. Finally, we consider the subcritical complex Ginzburg{Landau equation subject to a modi ed Pyragas control scheme, which includes a spatial feedback term. We nd that traveling wave solutions of the system cannot be stabilized with either spatial or temporal feedback.