# American Institute of Mathematical Sciences

August  2015, 35(8): 3417-3436. doi: 10.3934/dcds.2015.35.3417

## 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, South Korea, South Korea

Received  November 2014 Revised  January 2015 Published  February 2015

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.
Citation: Seung-Yeal Ha, Jinyeong Park, Sang Woo Ryoo. Emergence of phase-locked states for the Winfree model in a large coupling regime. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3417-3436. doi: 10.3934/dcds.2015.35.3417
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