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On the number of stable solutions in the Kuramoto model

  • Identification data

    Identifier: imarina:9330276
    Authors:
    Arenas, AlexGarijo, AntonioGomez, SergioVilladelprat, Jordi
    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.
  • Others:

    Author, as appears in the article.: Arenas, Alex; Garijo, Antonio; Gomez, Sergio; Villadelprat, Jordi
    Department: Enginyeria Informàtica i Matemàtiques
    URV's Author/s: Arenas Moreno, Alejandro / Garijo Real, Antonio / Gómez Jiménez, Sergio / Villadelprat Yagüe, Jordi
    Project code: PID2020-118281GB-C33
    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.
    Thematic Areas: Statistical and nonlinear physics Physics, mathematical Physics and astronomy (miscellaneous) Physics and astronomy (all) Medicine (miscellaneous) Medicina veterinaria Medicina ii Mathematics, applied Mathematical physics Matemática / probabilidade e estatística Interdisciplinar Geociências General physics and astronomy Engenharias iv Engenharias iii Engenharias ii Engenharias i Ciências ambientais Ciência da computação Astronomia / física Applied mathematics
    licence for use: https://creativecommons.org/licenses/by/3.0/es/
    Author's mail: sergio.gomez@urv.cat antonio.garijo@urv.cat alexandre.arenas@urv.cat
    Author identifier: 0000-0003-1820-0062 0000-0002-1503-7514 0000-0003-0937-0334
    Record's date: 2024-08-03
    Papper version: info:eu-repo/semantics/publishedVersion
    Link to the original source: https://pubs.aip.org/aip/cha/article/33/9/093127/2911850/On-the-number-of-stable-solutions-in-the-Kuramoto
    Funding program: Herramientas para el análisis de diagramas de bifurcación en sistemas dinámicos
    Papper original source: Chaos. 33 (9): 093127-
    APA: Arenas, Alex; Garijo, Antonio; Gomez, Sergio; Villadelprat, Jordi (2023). On the number of stable solutions in the Kuramoto model. Chaos, 33(9), 093127-. DOI: 10.1063/5.0161977
    Licence document URL: https://repositori.urv.cat/ca/proteccio-de-dades/
    Acronym: ATBiD
    Article's DOI: 10.1063/5.0161977
    Entity: Universitat Rovira i Virgili
    Journal publication year: 2023
    Funding program action: Proyectos I+D Generación de Conocimiento
    Publication Type: Journal Publications
  • Keywords:

    Applied Mathematics,Mathematical Physics,Mathematics, Applied,Medicine (Miscellaneous),Physics and Astronomy (Miscellaneous),Physics, Mathematical,Statistical and Nonlinear Physics
    Statistical and nonlinear physics
    Physics, mathematical
    Physics and astronomy (miscellaneous)
    Physics and astronomy (all)
    Medicine (miscellaneous)
    Medicina veterinaria
    Medicina ii
    Mathematics, applied
    Mathematical physics
    Matemática / probabilidade e estatística
    Interdisciplinar
    Geociências
    General physics and astronomy
    Engenharias iv
    Engenharias iii
    Engenharias ii
    Engenharias i
    Ciências ambientais
    Ciência da computação
    Astronomia / física
    Applied mathematics
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