Coupled Networks, Patterns and Complexity - Abstract
Schneider, Isabelle
We consider symmetrically coupled oscillators in Hopf normal form. We focus on the case of three oscillators as the simplest example admitting nontrivial symmetry. Our aim is to stabilize the inherently unstable periodic orbits with discrete rotating wave symmetry. These orbits emerge from a symmetric Hopf-bifurcation. Time-delayed feedback control (Pyragas control) is introduced. Going beyond previous results by Fiedler et al., we use the spatio-temporal symmetry of the system to establish a noninvasive control method. For that purpose we introduce symmetry-adapted coordinates. Due to a suitable control term the symmetric Hopf bifurcation splits into simple Hopf bifurcations with a two dimensional central manifold. As a consequence standard exchange of stability in two dimensions is now applicable. As a result, we are for the first time able to state explicit analytic conditions for the stabilization of three oscillators. These conditions are necessary and sufficient. We distinguish between the sub-and supercritical cases and within the subcritical case between hard and soft springs.