Delayed feedback stabilization with and without symmetry

Abstract

This thesis is concerned with delayed feedback stabilization of periodic orbits. Pyragas introduced a feedback scheme (now known as ‘Pyragas control’) that measures the difference between the current state and the state time p ago, and feeds the result back into the system. For p-periodic orbits, the difference between the current state and the state time p ago is zero; hence any p-periodic solution of the system without feedback is also a solution of the system with feedback. But the introduction of the feedback term changes the dynamics of the system, and periodic orbits that are unstable in the system without feedback can become stable in the system with feedback, as has been attestedin many experiments. Equivariant control follows the spirit of Pyragas control, but adapts it to situations where a periodic orbit satisfies a known spatial-temporal pattern. Equivariant control feeds back the difference between the current state and a spatial-temporal transformation of the state, in such a way that the difference vanishes on the known spatial-temporal pattern. Equivariant control is mainly used in systems with symmetries, where the symmetry of the system induces known spatial-temporal patterns on the periodic orbit. The first part of this thesis addresses limitations to Pyragas control. The main novelty here is that the limitations follow from a new invariance principle, which gives a clear and unifying understanding why Pyragas control can fail to stabilize. In the second part of the thesis, we consider periodic orbits where the group of ‘genuine’ spatial-temporal symmetries (i.e. spatial-temporal symmetries that are not actually spatial symmetries) is cyclic. Such periodic orbits are called discrete waves. We prove sufficient conditions under which equivariant control can stabilize discrete waves; this positive stabilization result is applicable to a broad class of discrete waves and the necessary conditions are formulated in terms of accessible information on the uncontrolled system. A running theme in the thesis is that we actively use the symmetry of the system (if present) in the stability analysis. This ‘equivariant stability analysis’ is both crucial on a technical level and in our understanding of equivariant control: it clarifies the connection with limitations to Pyragas control, and shows why (and in which situations) equivariant control is able to overcome these limitations.