Phase Transitions and Chaos in Long-Range Models of Coupled Oscillators

TL;DR

This paper investigates chaos and phase transitions in long-range coupled oscillator models, revealing how the Kuramoto and HMF models relate and how frequency distribution influences system behavior.

Contribution

It establishes a connection between the Kuramoto and Hamiltonian Mean Field models, highlighting their limiting case relationship and the impact of frequency distribution on phase transition dynamics.

Findings

Numerical results support the link between Kuramoto and HMF models.

The shape of the phase transition depends on the natural frequency distribution.

Largest Lyapunov exponent behavior varies with frequency distribution.

Abstract

We study the chaotic behavior of the synchronization phase transition in the Kuramoto model. We discuss the relationship with analogous features found in the Hamiltonian Mean Field (HMF) model. Our numerical results support the connection between the two models, which can be considered as limiting cases (dissipative and conservative, respectively) of a more general dynamical system of damped-driven coupled pendula. We also show that, in the Kuramoto model, the shape of the phase transition and the largest Lyapunov exponent behavior are strongly dependent on the distribution of the natural frequencies.