Non-invasive stabilization of periodic orbits in $O_4$-symmetrically coupled Van der Pol oscillators

TL;DR

This paper extends Pyragas time delayed feedback control to stabilize periodic orbits in a system of $O_4$-symmetrically coupled Van der Pol oscillators, providing explicit stability domains and leveraging group theory for control design.

Contribution

It adapts non-invasive control methods to a new symmetric oscillator system and explicitly characterizes stability regions using group theoretic insights.

Findings

Successfully stabilizes periodic solutions in $O_4$-symmetric Van der Pol oscillators.

Provides explicit parameter domains for stability of periodic orbits.

Demonstrates the effectiveness of group-theoretic restrictions in control design.

Abstract

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.