Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs

Francesca Di Patti, Duccio Fanelli, Filippo Miele, Timoteo Carletti

Résultats de recherche: Contribution à un journal/une revueArticle

Résumé

A normal form approximation for the evolution of a reaction-diffusion system hosted on a directed graph is derived, in the vicinity of a supercritical Hopf bifurcation. Weak diffusive couplings are assumed to hold between adjacent nodes. Under this working assumption, a Complex Ginzburg–Landau equation (CGLE) is obtained, whose coefficients depend on the parameters of the model and the topological characteristics of the underlying network. The CGLE enables one to probe the stability of the synchronous oscillating solution, as displayed by the reaction-diffusion system above Hopf bifurcation. More specifically, conditions can be worked out for the onset of the symmetry breaking instability that eventually destroys the uniform oscillatory state. Numerical tests performed for the Brusselator model confirm the validity of the proposed theoretical scheme. Patterns recorded for the CGLE resemble closely those recovered upon integration of the original Brussellator dynamics.
langueAnglais
Pages447-456
journalCommunication in Nonlinear Science and Numerical Simulation
Volume56
Les DOIs
étatPublié - 16 août 2017

Empreinte digitale

Complex Ginzburg-Landau Equation
Hopf bifurcation
Ginzburg-Landau
Directed graphs
Directed Graph
Reaction-diffusion System
Hopf Bifurcation
Approximation
Oscillating Solutions
Symmetry Breaking
Normal Form
Probe
Adjacent
Coefficient
Vertex of a graph
Model

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    title = "Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs",
    abstract = "A normal form approximation for the evolution of a reaction-diffusion system hosted on a directed graph is derived, in the vicinity of a supercritical Hopf bifurcation. Weak diffusive couplings are assumed to hold between adjacent nodes. Under this working assumption, a Complex Ginzburg–Landau equation (CGLE) is obtained, whose coefficients depend on the parameters of the model and the topological characteristics of the underlying network. The CGLE enables one to probe the stability of the synchronous oscillating solution, as displayed by the reaction-diffusion system above Hopf bifurcation. More specifically, conditions can be worked out for the onset of the symmetry breaking instability that eventually destroys the uniform oscillatory state. Numerical tests performed for the Brusselator model confirm the validity of the proposed theoretical scheme. Patterns recorded for the CGLE resemble closely those recovered upon integration of the original Brussellator dynamics.",
    keywords = "Reaction-diffusion model, Complex Ginzburg–Landau equation, Pattern formation, Synchronization",
    author = "{Di Patti}, Francesca and Duccio Fanelli and Filippo Miele and Timoteo Carletti",
    year = "2017",
    month = "8",
    day = "16",
    doi = "https://doi.org/10.1016/j.cnsns.2017.08.012",
    language = "English",
    volume = "56",
    pages = "447--456",
    journal = "Communication in Nonlinear Science and Numerical Simulation",
    issn = "1007-5704",
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    Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs. / Di Patti, Francesca ; Fanelli, Duccio; Miele, Filippo; Carletti, Timoteo.

    Dans: Communication in Nonlinear Science and Numerical Simulation, Vol 56, 16.08.2017, p. 447-456.

    Résultats de recherche: Contribution à un journal/une revueArticle

    TY - JOUR

    T1 - Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs

    AU - Di Patti,Francesca

    AU - Fanelli,Duccio

    AU - Miele,Filippo

    AU - Carletti,Timoteo

    PY - 2017/8/16

    Y1 - 2017/8/16

    N2 - A normal form approximation for the evolution of a reaction-diffusion system hosted on a directed graph is derived, in the vicinity of a supercritical Hopf bifurcation. Weak diffusive couplings are assumed to hold between adjacent nodes. Under this working assumption, a Complex Ginzburg–Landau equation (CGLE) is obtained, whose coefficients depend on the parameters of the model and the topological characteristics of the underlying network. The CGLE enables one to probe the stability of the synchronous oscillating solution, as displayed by the reaction-diffusion system above Hopf bifurcation. More specifically, conditions can be worked out for the onset of the symmetry breaking instability that eventually destroys the uniform oscillatory state. Numerical tests performed for the Brusselator model confirm the validity of the proposed theoretical scheme. Patterns recorded for the CGLE resemble closely those recovered upon integration of the original Brussellator dynamics.

    AB - A normal form approximation for the evolution of a reaction-diffusion system hosted on a directed graph is derived, in the vicinity of a supercritical Hopf bifurcation. Weak diffusive couplings are assumed to hold between adjacent nodes. Under this working assumption, a Complex Ginzburg–Landau equation (CGLE) is obtained, whose coefficients depend on the parameters of the model and the topological characteristics of the underlying network. The CGLE enables one to probe the stability of the synchronous oscillating solution, as displayed by the reaction-diffusion system above Hopf bifurcation. More specifically, conditions can be worked out for the onset of the symmetry breaking instability that eventually destroys the uniform oscillatory state. Numerical tests performed for the Brusselator model confirm the validity of the proposed theoretical scheme. Patterns recorded for the CGLE resemble closely those recovered upon integration of the original Brussellator dynamics.

    KW - Reaction-diffusion model

    KW - Complex Ginzburg–Landau equation

    KW - Pattern formation

    KW - Synchronization

    U2 - https://doi.org/10.1016/j.cnsns.2017.08.012

    DO - https://doi.org/10.1016/j.cnsns.2017.08.012

    M3 - Article

    VL - 56

    SP - 447

    EP - 456

    JO - Communication in Nonlinear Science and Numerical Simulation

    T2 - Communication in Nonlinear Science and Numerical Simulation

    JF - Communication in Nonlinear Science and Numerical Simulation

    SN - 1007-5704

    ER -