Stability Diagram, Hysteresis, and Critical Time Delay and Frequency for the Kuramoto Model with Heterogeneous Interaction Delays

TL;DR

This paper analyzes the stability, bifurcations, and hysteresis phenomena in large Kuramoto oscillator systems with heterogeneous interaction delays, deriving stability diagrams and identifying conditions for bistability and synchronization.

Contribution

It provides a comprehensive stability diagram for systems with exponential delay distributions and explores how delay heterogeneity affects synchronization and hysteresis.

Findings

Hysteresis occurs at a codimension-two bifurcation point.

Critical delay and frequency thresholds for bistability are identified.

More homogeneous delays promote synchronization and reduce hysteresis.

Abstract

We investigate the dynamics of large, globally-coupled systems of Kuramoto oscillators with heterogeneous interaction delays. For the case of exponentially distributed time delays we derive the full stability diagram that describes the bifurcations in the system. Of particular interest is the onset of hysteresis where both the incoherent and partially synchronized states are stable for a range of coupling strengths -- this occurs at a codimension-two point at the intersection between a Hopf bifucration and saddle-node bifurcation of cycles. By studying this codimension-two point we find the full set of characteristic time delays and natural frequencies where bistability exists and identify the critical time delay and critical natural frequency below which bistability does not exist. Finally, we examine the dynamics of the more general system where time delays are drawn from a Gamma distribution, finding that more homogeneous time delay distributions tend to both promote the onset of synchronization and inhibit the presence of hysteresis.