Non-invasive stabilization of periodic orbits in $O_4$-symmetrically   coupled Van der Pol oscillators
  Pyragas time delayed feedback control has proven itself as an effective tool to non-invasively stabilize periodic solutions. In a number of publications, this method was adapted to equivariant settings and applied to stabilize branches of small periodic solutions in systems of symmetrically coupled Landau oscillators near a Hopf bifurcation point. The form of the control ensures the non-invasiveness property, hence reducing the problem to finding a set of the gain matrices, which would guarantee the stabilization. In this paper, we apply this method to a system of Van der Pol oscillators coupled in a cube-like configuration leading to $O_4$-equivariance. We discuss group theoretic restrictions which help to shape our choice of control. Furthermore, we explicitly describe the domains in the parameter space for which the periodic solutions are stable. 
Zalman Balanov, Edward Hooton, Wieslaw Krawcewicz, Dmitrii Rachinskii
math.OC math.DS