Existence and robustness of phase-locking in coupled Kuramoto oscillators under mean-field feedback

Alessio Franci (1) , Antoine Chaillet (1) , William Pasillas-Lépine (1)

1 L2S - Laboratoire des signaux et systèmes

Motivated by the recent development of Deep Brain Stimulation (DBS) for neurological diseases, we study a network of interconnected oscillators under the influence of mean-field feedback and analyze the robustness of its phase-locking with respect to general inputs. Under standard assumptions, this system can be reduced to a modified version of the Kuramoto model of coupled nonlinear oscillators. In the first part of the paper we present an analytical study on the existence of phase-locked solutions under generic interconnection and feedback configurations. In particular we show that, in general, no oscillating phaselocked solutions can co-exist with any non-zero proportional mean-field feedback. In the second part we prove some robustness properties of phase-locked solutions (namely total stability). Thisgeneral result allows in particular to justify the persistence of practically phase-locked states if sufficiently small feedback gains are applied, and to give explicit necessary conditions on the intensity of a desynchronizing mean-field feedback. Furthermore, the Lyapunov function used in the analysis provides a new characterization of the robust phase-locked configurations in the Kuramoto system with symmetric interconnections.

Mots clés: Phase-locking; Mean-field feedback; Kuramoto oscillators; Deep Brain Stimulation

Domaines: Automatique / Robotique

Alessio Franci, Antoine Chaillet, William Pasillas-Lépine. Existence and robustness of phase-locking in coupled Kuramoto oscillators under mean-field feedback. Automatica, 2011, 47, pp.1193-1202. ⟨10.1016/j.automatica.2011.03.003⟩. ⟨hal-00526066⟩