A Two-Frequency-Two-Coupling model of coupled oscillators

TL;DR

This paper introduces a Two-Frequency-Two-Coupling (TFTC) model of oscillators with bimodal distributions, revealing complex phase-locking behaviors and stability properties influenced by correlation between frequencies and coupling strengths.

Contribution

The study develops an exact analytical framework for the TFTC model, uncovering novel phase-locked states and stability differences based on correlation of disorder, supported by numerical simulations.

Findings

Correlated disorder leads to two subpopulation states: Lock-Lock or Lock-Drift.

Uncorrelated disorder results in four phase-locked subpopulations with periodic global dynamics.

Incoherence is unstable with correlated disorder but neutrally stable with uncorrelated disorder.

Abstract

We considered the phase coherence dynamics in a Two-Frequency and Two-Coupling (TFTC) model of coupled oscillators, where coupling strength and natural oscillator frequencies for individual oscillators may assume one of two values (positive/negative). The bimodal distributions for the coupling strengths and frequencies are either correlated or uncorrelated. To study how correlation affects phase coherence, we analyzed the TFTC model by means of numerical simulations and exact dimensional reduction methods allowing to study the collective dynamics in terms of local order parameters. The competition resulting from distributed coupling strengths and natural frequencies produces nontrivial dynamic states. For correlated disorder in frequencies and coupling strengths, we found that the entire oscillator population splits into two subpopulations, both phase-locked (Lock-Lock) or one phase-locked, and the other drifting (Lock-Drift), where the mean-fields of the sub-populations maintain a constant non-zero phase difference. For uncorrelated disorder, we found that the oscillator population may split into four phase-locked subpopulations, forming phase-locked pairs which are either mutually frequency-locked (Stable Lock-Lock-Lock-Lock) or drifting (Breathing Lock-Lock-Lock-Lock), thus resulting in a periodic motion of the global synchronization level. Finally, we found for both types of disorder that a state of Incoherence exists; however, for correlated coupling strengths and frequencies, incoherence is always unstable, whereas it is only neutrally stable for the uncorrelated case. Numerical simulations performed on the model show good agreement with the analytic predictions. The simplicity of the model promises that real-world systems can be found which display the dynamics induced by correlated/uncorrelated disorder.