Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control

TL;DR

This paper demonstrates that time-delayed feedback control can stabilize certain unstable periodic orbits in the Lorenz system, challenging previous beliefs and extending the applicability of Pyragus control to higher-dimensional systems.

Contribution

It shows that the stabilization mechanism identified for the Hopf normal form also applies to the Lorenz equations, providing a new strategy for choosing feedback gains in complex systems.

Findings

Time-delayed feedback stabilizes unstable Lorenz orbits.

The stabilization mechanism extends beyond the Hopf normal form.

A broad parameter range achieves stabilization despite high system complexity.

Abstract

For many years it was believed that an unstable periodic orbit with an odd number of real Floquet multipliers greater than unity cannot be stabilized by the time-delayed feedback control mechanism of Pyragus. A recent paper by Fiedler et al uses the normal form of a subcritical Hopf bifurcation to give a counterexample to this theorem. Using the Lorenz equations as an example, we demonstrate that the stabilization mechanism identified by Fiedler et al for the Hopf normal form can also apply to unstable periodic orbits created by subcritical Hopf bifurcations in higher-dimensional dynamical systems. Our analysis focuses on a particular codimension-two bifurcation that captures the stabilization mechanism in the Hopf normal form example, and we show that the same codimension-two bifurcation is present in the Lorenz equations with appropriately chosen Pyragus-type time-delayed feedback. This example suggests a possible strategy for choosing the feedback gain matrix in Pyragus control of unstable periodic orbits that arise from a subcritical Hopf bifurcation of a stable equilibrium. In particular, our choice of feedback gain matrix is informed by the Fiedler et al example, and it works over a broad range of parameters, despite the fact that a center-manifold reduction of the higher-dimensional problem does not lead to their model problem.