Emergent behaviors of the continuum thermodynamic Kuramoto model in a large coupling regime

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 $$$ p<\infty $$$).…”

“…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.…”

“…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.…”

“…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.…”