Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation

TL;DR

This paper demonstrates that Pyragas delayed feedback control can stabilize unstable periodic orbits near a subcritical Hopf bifurcation in high-dimensional systems, extending previous results and providing explicit parameter choices.

Contribution

It extends the stabilization results of Pyragas feedback to higher-dimensional systems near subcritical Hopf bifurcations, with explicit formulas for feedback parameters.

Findings

Pyragas feedback stabilizes UPOs near subcritical Hopf bifurcations.

Explicit formulas for feedback gain parameters are derived.

Stabilization involves a degenerate Hopf bifurcation induced by delay.

Abstract

We show that Pyragas delayed feedback control can stabilize an unstable periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of a stable equilibrium in an n-dimensional dynamical system. This extends results of Fiedler et al. [PRL 98, 114101 (2007)], who demonstrated that such feedback control can stabilize the UPO associated with a two-dimensional subcritical Hopf normal form. Pyragas feedback requires an appropriate choice of a feedback gain matrix for stabilization, as well as knowledge of the period of the targeted UPO. We apply feedback in the directions tangent to the two-dimensional center manifold. We parameterize the feedback gain by a modulus and a phase angle, and give explicit formulae for choosing these two parameters given the period of the UPO in a neighborhood of the bifurcation point. We show, first heuristically, and then rigorously by a center manifold reduction for delay differential equations, that the stabilization mechanism involves a highly degenerate Hopf bifurcation problem that is induced by the time-delayed feedback. When the feedback gain modulus reaches a threshold for stabilization, both of the genericity assumptions associated with a two-dimensional Hopf bifurcation are violated: the eigenvalues of the linearized problem do not cross the imaginary axis as the bifurcation parameter is varied, and the real part of the cubic coefficient of the normal form vanishes. Our analysis of this degenerate bifurcation problem reveals two qualitatively distinct cases when unfolded in a two-parameter plane. In each case, Pyragas-type feedback successfully stabilizes the branch of small-amplitude UPOs in a neighborhood of the original bifurcation point, provided that the phase angle satisfies a certain restriction.