American Institute of Mathematical Sciences

doi: 10.3934/krm.2021036
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Uniform-in-time continuum limit of the lattice Winfree model and emergent dynamics

 1 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826 2 Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea 3 Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea 4 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea

*Corresponding author: Myeongju Kang

Received  November 2020 Revised  August 2021 Early access November 2021

We study a uniform-in-time continuum limit of the lattice Winfree model(LWM) and its asymptotic dynamics which depends on system functions such as natural frequency function and coupling strength function. The continuum Winfree model(CWM) is an integro-differential equation for the temporal evolution of Winfree phase field. The LWM describes synchronous behavior of weakly coupled Winfree oscillators on a lattice lying in a compact region. For bounded measurable initial phase field, we establish a global well-posedness of classical solutions to the CWM under suitable assumptions on coupling function, and we also show that a classical solution to the CWM can be obtained as a $L^1$-limit of a sequence of lattice solutions. Moreover, in the presence of frustration effect, we show that stationary states and bump states can emerge from some admissible class of initial data in a large and intermediate coupling regimes, respectively. We also provide several numerical examples and compare them with analytical results.

Citation: Seung-Yeal Ha, Myeongju Kang, Bora Moon. Uniform-in-time continuum limit of the lattice Winfree model and emergent dynamics. Kinetic & Related Models, doi: 10.3934/krm.2021036
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References:
Special pair for $S(\theta)$ and $I(\theta)$
Positions of $\bar\theta$ and ${\overline\theta}^\infty$ for case of $(SI)(\theta) = -\sin\theta(1 +\cos\theta)$
Uniform $L^1$-stability
Point trajectory of $\theta_1$ and $\theta_2$ at $(0.5, 0.5)$
Comparison between stationary profile and natural frequency profile
$L^1$-stability
Point trajectory of $\theta_1$ and $\theta_2$ at $(0.5, 0.5)$
Comparison between steady state and natural frequency
$\frac{\theta(x, 20)}{20}$, two different view points
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