Synchronization of Kuramoto Oscillators in Dense Networks

Abstract

We study synchronization properties of systems of Kuramoto oscillators. The problem can also be understood as a question about the properties of an energy landscape created by a graph. More formally, let $G=(V,E)$ be a connected graph and $(a_{ij}{)}_{i,j=1}^n$ denotes its adjacency matrix. Let the function $f:{\mathbb{T}}^n\to \mathbb{R}$ be given by $$f({\theta }_1,\dots ,{\theta }_n)=\sum _{i,j=1}^n{a}_{ij}\cos ({\theta }_i-{\theta }_j).$$ This function has a global maximum when ${\theta }_i=\theta$ for all $1\le i\le n$. It is known that if every vertex is connected to at least $\mu (n-1)$ other vertices for $\mu$ sufficiently large, then every local maximum is global. Taylor proved this for $\mu \ge 0.9395$ and Ling, Xu \& Bandeira improved this to $\mu \ge 0.7929$. We give a slight improvement to $\mu \ge 0.7889$. Townsend, Stillman \& Strogatz suggested that the critical value might be ${\mu }_c=0.75$.

References in corpus (2)

• Algebraic Geometrization of the Kuramoto Model: Equilibria and Stability Analysis
• Stability of twisted states in the Kuramoto model on Cayley and random graphs

Cited by in corpus (6)

• A global synchronization theorem for oscillators on a random graph
• Sufficiently dense Kuramoto networks are globally synchronizing
• The lower bound of the network connectivity guaranteeing in-phase synchronization
• Benign landscapes of low-dimensional relaxations for orthogonal synchronization on general graphs
• Expander graphs are globally synchronizing
• Sparsity-driven synchronization in oscillators networks