The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations

Abstract

In our previous work [3], we initiated a mathematical investigation of the onset of synchronization in the Kuramoto model (KM) of coupled phase oscillators on convergent graph sequences. There, we derived and rigorously justified the mean field limit for the KM on graphs. Using linear stability analysis, we identified the critical values of the coupling strength, at which the incoherent state looses stability, thus, determining the onset of synchronization in this model.In the present paper, we study the corresponding bifurcations. Specifically, we show that similar to the original KM with all-to-all coupling, the onset of synchronization in the KM on graphs is realized via a pitchfork bifurcation. The formula for the stable branch of the bifurcating equilibria involves the principal eigenvalue and the corresponding eigenfunctions of the kernel operator defined by the limit of the graph sequence used in the model. This establishes an explicit link between the network structure and the onset of synchronization in the KM on graphs. The results of this work are illustrated with the bifurcation analysis of the KM on Erdős-Rényi, small-world, as well as certain weighted graphs on a circle.

Mean Field Limit for Stochastic Particle Systems

“…Random weights w ij are popular for this reason and naturally lead to random networks, see [4]. The corresponding graph of interactions is typically dense and some notion of mean-field limit, based on graphons, has been derived for example in [21,19,20] for smooth interaction kernels K, and actually for the so-called Kuramoto [50,51] model without learning (see (80) below with η = 0).…”

“…In this way, we associate to the discrete connectivities w ij a function w that accounts for the effect of the architecture of the complex original system (1). This type of graphon-like representation had previously been explored in particular in [19,20,21] and more recently in [37,48]. However those results required more stringent assumptions on the connectivities, so that [19,20,21] only apply to dense graphs or symmetric interactions, whilst [37,48] allow for sparse graphs but with a priori knowledge of some additional convergence of the coupling weights w ij .…”

“…This type of graphon-like representation had previously been explored in particular in [19,20,21] and more recently in [37,48]. However those results required more stringent assumptions on the connectivities, so that [19,20,21] only apply to dense graphs or symmetric interactions, whilst [37,48] allow for sparse graphs but with a priori knowledge of some additional convergence of the coupling weights w ij . The major contribution of the present article is to remove all such assumptions and only require the assumptions (3)-( 4).…”

“…then, there exists a measure-preserving map φ N on [0, 1] such that we can pass to the limit in f N (t, x, φ N (ξ)) to a solution f (t, x, ξ) to (5). As a consequence, we can determine the limit of the 1-particle distribution given by 1 0 f N (t, x, ξ) dξ: We refer to [19,20,21] in particular for a complete analysis that this brief sketch cannot do justice to.…”

Mathematical Framework for Breathing Chimera States

“…6b-d. Note that proper consideration of such bifurcations requires the use of generalized spectral methods (Chiba and Nishikawa 2011;Dietert 2016;Chiba and Medvedev 2019), which, however, must be adapted to a situation where the reference solution is a relative periodic orbit rather than a simple equilibrium.…”