Delay Stabilization of Rotating Waves without Odd Number Limitation

Abstract

A variety of methods have been developed in nonlinear science to stabilize unstable periodic orbits (UPOs) and control chaos [1], following the seminal work by Ott, Grebogi and Yorke [2], who employed a tiny control force to stabilize UPOs embedded in a chaotic attractor [3,4]. A particularly simple and efficient scheme is time-delayed feedback as suggested by Pyragas [5], which uses the difference z(t − τ) − z(t) of a signal z at a time t and a delayed time t − τ. It is an attempt to stabilize periodic orbits of (minimal) period T by a feedback control which involves a time delay τ = nT, for suitable positive integer n. A linear feedback example iṡ(2.1) whereż(t) = f (λ, z(t)) describes a d-dimensional nonlinear dynamical system with bifurcation parameter λ and an unstable orbit of (minimal) period T. The constant feedback control matrix B is chosen suitably. Typical choices are multiples of the identity or of rotations, or matrices of low rank. More general nonlinear feedbacks are conceivable, of course. The main point, however, is that the Pyragas choice τ P

Control of Complex Nonlinear Systems with Delay

“…[FIE07,FIE07a,FIE07b]. This counterexample consists of the normal form of a subcritical Hopf bifurcation subject to time-delayed feedback control.…”

“…(4.2.1) by Pyragas control following Refs. [FIE07b,JUS07]. Specifically, I will address S 1 -equivariance and co-rotating coordinates z(t) = e iωt ζ(t), rotating waves and their bifurcations, the role of center manifolds, and bifurcation to non-rotating waves.…”