Noninvasive Stabilization of Periodic Orbits in O4-Symmetrically Coupled Systems Near a Hopf Bifurcation Point

Abstract

Pyragas time-delayed feedback control has proven itself as an effective tool to noninvasively stabilize periodic solutions. In a number of publications, this method was adapted to equivariant settings. In this paper, we consider [Formula: see text]-symmetric systems of van der Pol and optical oscillators coupled in a cube-like configuration. These systems undergo equivariant Hopf bifurcations giving rise to multiple branches of unstable periodic solutions. We introduce a delayed control term, which ensures stabilization of a selected branch. Group theoretic restrictions which help to shape our choice of control are discussed. Furthermore, we explicitly describe the domains in a two-dimensional parameter space for which the periodic solutions of the delayed system are stable.