Emergence of phase-locked states for the Winfree model in a large coupling regime

Seung-Yeal Ha; Jinyeong Park; Sang Woo Ryoo

doi:10.3934/dcds.2015.35.3417

Article Contents

2015, Volume 35, Issue 8: 3417-3436. Doi: 10.3934/dcds.2015.35.3417

This issue Previous Article Multi-bump solutions for Schrödinger equation involving critical growth and potential wells Next Article Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows

Emergence of phase-locked states for the Winfree model in a large coupling regime

1.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747
2.
Department of Mathematical Sciences, Seoul National University, Seoul, 151-747

Received: October 31, 2014

Revised: December 31, 2014

Published: May 2017

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Abstract

We study the large-time behavior of the globally coupled Winfree model in a large coupling regime. The Winfree model is the first mathematical model for the synchronization phenomenon in an ensemble of weakly coupled limit-cycle oscillators. For the dynamic formation of phase-locked states, we provide a sufficient framework in terms of geometric conditions on the coupling functions and coupling strength. We show that in the proposed framework, the emergent phase-locked state is the unique equilibrium state and it is asymptotically stable in an $l^1$-norm; further, we investigate its configurational structure. We also provide several numerical simulations, and compare them with our analytical results.

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Mathematics Subject Classification: Primary: 70F99, 92B25.

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