Existence of phase-locking in the Kuramoto system under mean-fi eld feedback

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

1 L2S - Laboratoire des signaux et systèmes

Motivated by the development of Deep Brain Stimulation (DBS) for neurological diseases, we study a network of interconnected oscillators under the influence of a proportional mean-fi eld feedback. Under standard assumptions, this system can be reduced to a modi fied version of the Kuramoto model of coupled nonlinear oscillators. In the first part of the paper we show that, in general, no oscillating phase-locked solution can co-exist with any non-zero feedback gain. In the second part we propose a new characterization of phase-locking between Kuramoto oscillators. In particular we derive a fi xed point equation for the Kuramoto system under mean- field feedback and we show how, generically, the "standard" (with zero feedback gain) Kuramoto fixed point equation is locally invertible in terms of the implicit function theorem.

Mots clés: Kuramoto Oscillators; Mean- eld Feedback; Phase-Locking; Deep Brain Stimulation

Domaines: Automatique / Robotique

Alessio Franci, William Pasillas-Lépine, Antoine Chaillet. Existence of phase-locking in the Kuramoto system under mean-fi eld feedback. IFAC World Congress, Aug 2011, Milan, Italy. pp.1-6. ⟨hal-00652617⟩