Global Effects of Time-Delayed Feedback Control Applied to the Lorenz System

Abstract

Time-delayed feedback control was introduced by Pyragas in 1992 as a general method for stabilizing an unstable periodic orbit of a given continuous-time dynamical system. The analysis of Pyragas control focused on its application to the normal form of a subcritical Hopf bifurcation, and it was initially concerned with stabilization near the Hopf bifurcation. A recent study considered this normal form delay differential equation model more globally in terms of its bifurcation structure for any values of system and control parameters. This revealed families of delay-induced Hopf bifurcations and secondary stability regions of periodic orbits. In this contribution we show that these results for the normal form are relevant in an application context. To this end, we present a case study of the well-known Lorenz system subject to Pyragas control to stabilize one of its two (symmetrically related) saddle periodic orbits. We find that, for a suitably chosen value of the \(2\pi \)-periodic feedback phase, the controlled Hopf normal form describes qualitatively the bifurcation set and relevant stability regions that exist in the controlled Lorenz system down to the homoclinic bifurcation where the target saddle periodic orbit is born. In particular, there are secondary stability regions of periodic orbits in the Lorenz system. Finally, the normal form also describes correctly the effect of a delay mismatch.