# Onset of synchronization in networks of second-order Kuramoto oscillators with delayed coupling: Exact results and application to phase-locked loops

## TL;DR

This paper analyzes how time delay and inertia influence the onset of synchronization in a network of second-order Kuramoto oscillators, providing exact results and applying them to phase-locked loops.

## Contribution

It offers an exact theoretical analysis of the bifurcation behavior in delayed second-order Kuramoto models, revealing the conditions for subcritical and supercritical transitions.

## Key findings

- Delay affects the nature of the synchronization bifurcation.
- Inertia and delay determine whether the transition is subcritical or supercritical.
- Theoretical predictions are validated by numerical simulations.

## Abstract

We consider the inertial Kuramoto model of $N$ globally coupled oscillators characterized by both their phase and angular velocity, in which there is a time delay in the interaction between the oscillators. Besides the academic interest, we show that the model can be related to a network of phase-locked loops widely used in electronic circuits for generating a stable frequency at multiples of an input frequency. We study the model for a generic choice of the natural frequency distribution of the oscillators, to elucidate how a synchronized phase bifurcates from an incoherent phase as the coupling constant between the oscillators is tuned. We show that in contrast to the case with no delay, here the system in the stationary state may exhibit either a subcritical or a supercritical bifurcation between a synchronized and an incoherent phase, which is dictated by the value of the delay present in the interaction and the precise value of inertia of the oscillators. Our theoretical analysis, performed in the limit $N \to \infty$, is based on an unstable manifold expansion in the vicinity of the bifurcation, which we apply to the kinetic equation satisfied by the single-oscillator distribution function. We check our results by performing direct numerical integration of the dynamics for large $N$, and highlight the subtleties arising from having a finite number of oscillators.