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The Kuramoto Model: The Stability Conditions in the Presence of Phase Shift

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Abstract

A set of coupled Kuramoto oscillators is the main applied model for harmonization study of oscillating phenomena in physical, biological and engineering networks. In line with the previous studies and to bring the analytical results into conformity with further realistic models, in present paper the synchronization of Kuramoto oscillators has been investigated and the necessary and sufficient conditions for the frequency synchronization and phase cohesiveness have been introduced using the contraction property. The novelty of this paper lies in the following: (I) we consider the heterogeneous second-order model with non-uniformity in coupling topology; (II) we apply a non-zero and non-uniform phase shift in coupling function; (III) we introduce a new Lyapunov-based stability analysis technique. We demonstrate how the heterogeneity in network and the phase shift in coupling function are the key factors in network synchronization. The synchronization conditions are presented on the basis of network graph-theoretical characteristics and the oscillators’ parameters. Investigation of the analytical results reveals that an increase in the phase shift and heterogeneity of oscillators will complicate the synchronization conditions. The validity of the main theoretical results has been confirmed through the numerical simulations.

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Abbreviations

RHS:

Right hand side

\(M_{i}\) :

Rotational inertia in oscillator \(i\)

\(\theta_{i}\) :

ith oscillator phase

\(\dot{\theta }_{i}\) :

Frequency deviation

\(D_{i}\) :

Damping coefficient in oscillator \(i\)

\(\omega_{i}\) :

Input power or natural frequency in oscillator \(i\)

\(P_{ij}\) :

Coupling strength

\(\phi_{ij}\) :

Phase shift parameter

\(H\overline{\vartheta }\) :

Parameter related to \(H\theta_{ij}^{0}\) and \(\phi_{ij}\)

\(\theta_{e}\) :

Network equilibrium point

\(P^{0}\) :

Initial coupling strength before transient

\(P_{c}\) :

Critical coupling strength

\(\omega_{syn}\) :

Synchronized frequency

\(\omega_{syn}^{0}\) :

Synchronized frequency at \(t = 0\)

\(\phi_{cri}\) :

Critical value of phase shift

LHS:

Left hand side

\(\theta_{i}^{0}\) :

ith oscillator initial phase

\(\rho\) :

Parameter related to phase difference

\(B = H^{T}\) :

Incidence matrix

\(A\) :

Adjacency matrix

\(G\) :

Weighted directed graph

\({\mathbb{T}}^{1}\) :

1-Torus

\(\Delta\) :

An close set

\(\varepsilon\) :

Sets of vertices

\(v\) :

Sets of edges

\(c\) :

Coefficient

\(V\) :

Lyapunov function

\(\mu_{c}\) :

Critical angle

\(\omega_{Dif} ,X,X_{U} ,X_{L} , S_{\omega } ,{\Upsilon ,}K_{suf}\) :

Computational parameters or vectors based on network parameters

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This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Authors and Affiliations

  1. The Department of Electrical and Computer Engineering, Tarbiat Modares University, P.O. Box 14115-111, Tehran, Iran

    Reza Farhangi & Mohammad Taghi Hamidi Beheshti

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  1. Reza Farhangi
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  2. Mohammad Taghi Hamidi Beheshti
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Correspondence to Reza Farhangi.

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Farhangi, R., Hamidi Beheshti, M.T. The Kuramoto Model: The Stability Conditions in the Presence of Phase Shift. Neural Process Lett 53, 2631–2648 (2021). https://doi.org/10.1007/s11063-021-10510-0

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