Low-Dimensional Behavior of a Kuramoto Model with Inertia and Hebbian Learning

TL;DR

This paper investigates the low-dimensional dynamics of a Kuramoto model with inertia and Hebbian learning, analyzing stability and behaviors of coupled oscillators, and extending insights to high-dimensional systems with Gaussian-distributed frequencies.

Contribution

It provides an exact solution for the longitudinal dynamics and a detailed stability analysis of the transverse system in a simplified two-oscillator case, linking to high-dimensional behavior.

Findings

Transverse system trajectories are confined to a bounded region.

Identified three qualitatively different behaviors based on parameter regions.

Extended analysis to high-dimensional systems with Gaussian frequency distributions.

Abstract

We study low-dimensional dynamics in a Kuramoto model with inertia and Hebbian learning. In this model, the coupling strength between oscillators depends on the phase differences between the oscillators and changes according to a Hebbian learning rule. We analyze the special case of two coupled oscillators, which yields a five-dimensional dynamical system that decouples into a two-dimensional longitudinal system and a three-dimensional transverse system. We readily write an exact solution of the longitudinal system, and we then focus our attention on the transverse system. We classify the stability of the transverse system's equilibrium points using linear stability analysis. We show that the transverse system is dissipative and that all of its trajectories are eventually confined to a bounded region. We compute Lyapunov exponents to infer the transverse system's possible limiting behaviors, and we demarcate the parameter regions of three qualitatively different behaviors. Using insights from our analysis of the low-dimensional dynamics, we study the original high-dimensional system in a situation in which we draw the intrinsic frequencies of the oscillators from Gaussian distributions with different variances.