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

Alessio Franci 1, * William Pasillas-Lépine 1 Antoine Chaillet 1
* Corresponding author
1 Division Systèmes - L2S
L2S - Laboratoire des signaux et systèmes : 1289
Abstract : 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.
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Submitted on : Thursday, December 15, 2011 - 11:15:38 PM
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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⟩

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