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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

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

Pages: 3417 - 3436, Volume 35, Issue 8, August 2015      doi:10.3934/dcds.2015.35.3417

 
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Seung-Yeal Ha - Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747, South Korea (email)
Jinyeong Park - Department of Mathematical Sciences, Seoul National University, Seoul, 151-747, South Korea (email)
Sang Woo Ryoo - Department of Mathematical Sciences, Seoul National University, Seoul, 151-747, South Korea (email)

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.

Keywords:  Phase model, phase-locked state, synchronization, Winfree model, emergence.
Mathematics Subject Classification:  Primary: 70F99, 92B25.

Received: November 2014;      Revised: January 2015;      Available Online: February 2015.

 References