# Stochastic phase-cohesiveness of discrete-time Kuramoto oscillators in a frequency-dependent tree network

## TL;DR

This paper introduces stochastic phase-cohesiveness for discrete-time Kuramoto oscillators in frequency-dependent tree networks, providing conditions for synchronization under uncertainty using stochastic systems analysis.

## Contribution

It develops a novel stochastic phase-cohesiveness concept for Kuramoto models with frequency uncertainties on tree networks, deriving conditions for synchronization.

## Key findings

- Derived sufficient coupling strength condition for phase-cohesiveness.
- Established necessary sampling-period condition for stochastic synchronization.
- Validated theoretical results through numerical simulations.

## Abstract

This paper presents the notion of stochastic phase-cohesiveness based on the concept of recurrent Markov chains and studies the conditions under which a discrete-time stochastic Kuramoto model is phase-cohesive. It is assumed that the exogenous frequencies of the oscillators are combined with random variables representing uncertainties. A bidirectional tree network is considered such that each oscillator is coupled to its neighbors with a coupling law which depends on its own noisy exogenous frequency. In addition, an undirected tree network is studied. For both cases, a sufficient condition for the common coupling strength and a necessary condition for the sampling-period are derived such that the stochastic phase-cohesiveness is achieved. The analysis is performed within the stochastic systems framework and validated by means of numerical simulations.