Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model

Abstract

The existence and uniqueness/multiplicity of phase locked solution for continuum Kuramoto model was studied in [12, 29]. However, its asymptotic behavior is still unknown. In this paper we concern the asymptotic property of classic solutions to continuum Kuramoto model. In particular, we prove the convergence towards a phase locked state and its stability, provided suitable initial data and coupling strength. The main strategy is the quasi-gradient flow approach based on Lojasiewicz inequality. For this aim, we establish a Lojasiewicz type inequality in infinite dimensions for continuum Kuramoto model which is a nonlocal integro-differential equation. General theorems for convergence and stability of (generalized) quasi-gradient system in an abstract setting are also provided based on Lojasiewicz inequality.

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

“…Next, we show that the Cauchy problem (2) admits a unique global classical solution in the sense of Definition 3.1 using the mild formulation of (2) and a contraction mapping principle on a suitable complete metric space. Similar arguments for the continuum Kuramoto model can be found in [38,41]. First, we note that equation (2) 1 can be rewritten as a mild form:…”

“…A rigorous mathematical justification for the continuum limit of the Kuramoto model on networks with increasing number of nodes has been studied by combining the theory of evolution equations and graph limit [40,41,42]. Recently, asymptotic stability analysis of the continuum Kuramoto model has been studied in [38] using Lojasiewicz inequality and gradient flow approach. To put the LWM in a proper framework, we begin with a suitable set of jargons.…”

“…In this way, we can extend a local solution to any time interval. Detailed arguments can be found in [38,41].…”