Uniform-in-time continuum limit of the lattice Winfree model and emergent dynamics

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula>-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.</p>

Continuum limit of the lattice Lohe group model and emergent dynamics

“…First, we employed a communication function compared to the original Lohe group model 15 . Second, we adopted a compact region in $$$ d $$$‐dimensional Euclidean space whose boundary has an upper box dimension strictly less than $$$ d $$$ so that we can use supremum norm in the continuum limit of the lattice Lohe group model (see literature 25,26,28,29 in $$$ {L}^p $$$ setting with $$$ p<\infty $$$).…”

“…Definition 5.1 (Literature 25,26,28 ). For given 𝜏 ∈ (0, ∞), we say the continuum model (1.3) is "derivable" from the lattice model (1.2) in [0, 𝜏], if a solution to (1.3) can be obtained as a suitable limit of a sequence of lattice solutions to (1.2), as the lattice spacing tends to zero.…”

“…Recently, the above questions have been treated for low-dimensional models such as the Winfree model 25,26 and the Kuramoto model. [27][28][29] Next, we summarize our main results.…”

“…To the best of our knowledge, it is well known that the limiting process from lattice model to continuum model in $$$ {L}^p $$$ sense for finite $$$ p $$$ can be justified rigorously on finite time interval by using the Lebesgue differentiation theorem, and it can even be done uniformly in time. We refer to literature 25,26,28,29 for details. However, we prove this convergence in supremum norm sense, which is the complete norm of the solution space.…”

Emergence of phase-locked states for a deterministic and stochastic Winfree model with inertia

“…In [19,22], it is proven that phase-locked states emerge for sufficiently small κ < |ν| and under suitable restrictions on the spread ν i − ν j and θ i 0 − θ j 0 of the natural frequencies and the initial data. Further works on Winfree-type models include results on continuum limits [14], adaptive couplings [17], and models with time-delay [16] and frustration [13].…”