A Global Bifurcation Analysis of the Subcritical Hopf Normal Form Subject to Pyragas Time-Delayed Feedback Control

Unstable periodic orbits occur naturally in many nonlinear dynamical systems. They can generally not be observed directly, but a number of control schemes have been suggested to stabilize them. One such scheme is that by Pyragas [Phys. Lett. A, 170 (1992), pp. 421-428], which uses time-delayed feedback to target a specific unstable periodic orbit of a given period and stabilize it. This paper considers the global effect of applying Pyragas control to a nonlinear dynamical system. Specifically, we consider the standard example of the subcritical Hopf normal form subject to Pyragas control, which is a delay differential equation (DDE) that models how a generic unstable periodic orbit is stabilized. Our aim is to study how this DDE model depends on its different parameters, including the phase of the feedback and the imaginary part of the cubic coefficient, over their entire ranges. We show that the delayed feedback control induces infinitely many curves of Hopf bifurcations, from which emanate infinitely many periodic orbits that, in turn, have further bifurcations. Moreover, we show that, in addition to the stabilized target periodic orbit, there are possibly infinitely many stable periodic orbits. We compactify the parameter plane to show how these Hopf bifurcation curves change when the 2 pi-periodic phase of the feedback is varied. In particular, the domain of stability of the target periodic orbit changes in this process, and, at certain parameter values, it disappears completely. Overall, we present a comprehensive global picture of the dynamics induced by Pyragas control.