Stabilisation of long-period periodic orbits using time-delayed feedback control

TL;DR

This paper demonstrates that a modified time-delayed feedback control method can stabilize long-period periodic orbits arising from resonant bifurcations of heteroclinic cycles, with analytical and numerical validation.

Contribution

It introduces a new approach to stabilize arbitrarily large period orbits using a simplified three-dimensional map derived from delay equations.

Findings

Analytical stability conditions derived for the control method.

Good agreement between analytical predictions and numerical simulations.

Applicable to orbits from resonant bifurcations of heteroclinic cycles.

Abstract

The Pyragas method of feedback control has attracted much interest as a method of stabilising unstable periodic orbits in a number of situations. We show that a time-delayed feedback control similar to the Pyragas method can be used to stabilise periodic orbits with arbitrarily large period, specifically those resulting from a resonant bifurcation of a heteroclinic cycle. Our analysis reduces the infinite-dimensional delay-equation governing the system with feedback to a three-dimensional map, by making certain assumptions about the form of the solutions. The stability of a fixed point in this map corresponds to the stability of the periodic orbit in the flow, and can be computed analytically. We compare the analytic results to a numerical example and find very good agreement.