Affiliation:
1. Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili 1 , 43007 Tarragona, Spain
2. Pacific Northwest National Laboratory 2 , 902 Battelle Blvd, Richland, Washington 99354, USA
Abstract
We consider a system of n coupled oscillators described by the Kuramoto model with the dynamics given by θ˙=ω+Kf(θ). In this system, an equilibrium solution θ∗ is considered stable when ω+Kf(θ∗)=0, and the Jacobian matrix Df(θ∗) has a simple eigenvalue of zero, indicating the presence of a direction in which the oscillators can adjust their phases. Additionally, the remaining eigenvalues of Df(θ∗) are negative, indicating stability in orthogonal directions. A crucial constraint imposed on the equilibrium solution is that |Γ(θ∗)|≤π, where |Γ(θ∗)| represents the length of the shortest arc on the unit circle that contains the equilibrium solution θ∗. We provide a proof that there exists a unique solution satisfying the aforementioned stability criteria. This analysis enhances our understanding of the stability and uniqueness of these solutions, offering valuable insights into the dynamics of coupled oscillators in this system.
Funder
Ministerio de Ciencia e Innovación
Agència de Gestió d'Ajuts Universitaris i de Recerca
James S. McDonnell Foundation
Universitat Rovira i Virgili
Institució Catalana de Recerca i Estudis Avançats
Pacific Northwest National Laboratory
HORIZON EUROPE Framework Programme
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics