Stochastic Stability of Discrete-time Phase-coupled Oscillators over Uncertain and Random Networks

Journal article (2023)

Authors

M. Jafarian Team Matin Jafarian - Mechanical, Maritime and Materials Engineering
Mohammad H. Mamduhi ETH Zürich
Karl Henrik Johansson KTH Royal Institute of Technology
Research Group
Team Matin Jafarian (Mechanical, Maritime and Materials Engineering) (TU Delft)
Synchronization Oscillators Markov processes Uncertainty Stability analysis Stochastic processes Nonlinear systems Couplings Stochastic systems
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Published Date

2023

Language

English

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Faculty

Mechanical, Maritime and Materials Engineering

Department

Delft Center for Systems and Control

Research Group

Team Matin Jafarian

Abstract

This article studies stochastic relative phase stability, i.e., stochastic phase-cohesiveness, of discrete-time phase-coupled oscillators. The stochastic phase-cohesiveness in two types of networks is studied. First, we consider oscillators coupled with 2π-periodic odd functions over underlying undirected graphs subject to both multiplicative and additive stochastic uncertainties. We prove stochastic phase-cohesiveness of the network with respect to two specific, namely, in-phase and antiphase, sets by deriving sufficient coupling conditions. We show the dependency of these conditions on the size of the mean values of additive and multiplicative uncertainties, as well as the sign of the mean values of multiplicative uncertainties. Furthermore, we discuss the results under a relaxation of the odd property of the coupling function. Second, we study an uncertain network in which the multiplicative uncertainties are governed by the Bernoulli process representing the well-known Erdös-Rényi network. We assume constant exogenous frequencies and derive sufficient conditions for achieving both stochastic phase-cohesive and phase-locked solutions, i.e., stochastic phase-cohesiveness with respect to the origin. For the latter case, where identical exogenous frequencies are assumed, we prove that any positive probability of connectivity leads to phase-locking. Thorough analyses are provided, and insights obtained from stochastic analysis are discussed, along with numerical simulations to validate the analytical results.