Stability and Synchronization of Kuramoto Oscillators

TL;DR

This work addresses the stability and synchronization of Kuramoto oscillators on graphs by adopting a graph-theoretic formulation with incidence matrix $B$ and Laplacian $L$, yielding the dynamics $\dot{\theta} = \omega - \frac{K}{N} B \sin(B^T \theta)$. It derives fixed-point existence and uniqueness criteria, presents bounds on the coupling strength $K$ for synchronization, and introduces notions of phase cohesiveness and onset of synchronization, including exponential convergence rates tied to the weighted Laplacian $L_W$. The paper further applies these results to practical settings such as power-grid models, coupled oscillator networks with springs, and Vicsek-inspired vehicle coordination, supplemented by simulations and visualizations (order parameter dynamics, Manim animations). Overall, it provides a rigorous, graph-based toolkit for analyzing when and how networks of Kuramoto oscillators achieve synchronized behavior, with direct implications for engineering and biological systems. The combination of Lyapunov-based arguments, fixed-point theory, and spectral graph theory offers both theoretical guarantees and actionable design criteria for achieving robust synchronization in complex networks.

Abstract

Imagine a group of oscillators, each endowed with their own rhythm or frequency, be it the ticking of a biological clock, the swing of a pendulum, or the glowing of fireflies. While these individual oscillators may seem independent of one another at first glance, the true magic lies in their ability to influence and synchronize with one another, like a group of fireflies glowing in unison. The Kuramoto model was motivated by this phenomenon of collective synchronization, when a group of a large number of oscillators spontaneously lock to a common frequency, despite vast differences in their individual frequencies. Inspired by Kuramoto's groundbreaking work in the 1970s, this model captures the essence of how interconnected systems, ranging from biological networks to power grids, can achieve a state of synchronization. This work aims to study the stability and synchronization of Kuramoto oscillators, starting off with an introduction to Kuramoto Oscillators and it's broader applications. We then at a graph theoretic formulation for the same and establish various criterion for the stability, synchronization of Kuramoto Oscillators. Finally, we broadly analyze and experiment with various physical systems that tend to behave like Kuramoto oscillators followed by further simulations.

Figures (11)

• Figure 1: Evolution of $r(t)$
• Figure 2: for a 2-node generator-load net
• Figure 3: Springs on a ring
• Figure 4: Evolution of $r(t)$ with time
• Figure 5: Showing Phase Coherence for increasing $K$