The existence and classification of synchrony-breaking bifurcations in regular homogeneous networks using lattice structures

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    Abstract

    For regular homogeneous networks with simple eigenvalues (real or complex), all possible explicit forms of lattices of balanced equivalence relations can be constructed by introducing lattice generators and lattice indices [Kamei, 2009]. Balanced equivalence relations in the lattice correspond to clusters of partially synchronized cells in a network. In this paper, we restrict attention to regular homogeneous networks with simple real eigenvalues, and one-dimensional internal dynamics for each cell. We first show that lattice elements with nonzero indices indicate the existence of codimension-one synchrony-breaking steady-state bifurcations, and furthermore, the positions of such lattice elements give the number of partially synchronized clusters. Using four-cell regular homogeneous networks as an example, we then classify a large number of regular homogeneous networks into a small number of lattice structures, in which networks share an equivalent clustering type. Indeed, some of these networks even share the same generic bifurcation structure. This classification leads us to explore how regular homogeneous networks that share synchrony-breaking bifurcation structure are topologically related.
    Original languageEnglish
    Pages (from-to)3707-3732
    Number of pages26
    JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
    Volume19
    Issue number11
    DOIs
    Publication statusPublished - Nov 2009

    Fingerprint

    Synchrony
    Lattice Structure
    Network Structure
    Bifurcation
    Equivalence relation
    Cell
    Eigenvalue
    Codimension
    Classify
    Clustering
    Generator
    Internal

    Keywords

    • Coupled cell networks
    • Lattice
    • Synchrony breaking bifurcation

    Cite this

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    title = "The existence and classification of synchrony-breaking bifurcations in regular homogeneous networks using lattice structures",
    abstract = "For regular homogeneous networks with simple eigenvalues (real or complex), all possible explicit forms of lattices of balanced equivalence relations can be constructed by introducing lattice generators and lattice indices [Kamei, 2009]. Balanced equivalence relations in the lattice correspond to clusters of partially synchronized cells in a network. In this paper, we restrict attention to regular homogeneous networks with simple real eigenvalues, and one-dimensional internal dynamics for each cell. We first show that lattice elements with nonzero indices indicate the existence of codimension-one synchrony-breaking steady-state bifurcations, and furthermore, the positions of such lattice elements give the number of partially synchronized clusters. Using four-cell regular homogeneous networks as an example, we then classify a large number of regular homogeneous networks into a small number of lattice structures, in which networks share an equivalent clustering type. Indeed, some of these networks even share the same generic bifurcation structure. This classification leads us to explore how regular homogeneous networks that share synchrony-breaking bifurcation structure are topologically related.",
    keywords = "Coupled cell networks, Lattice, Synchrony breaking bifurcation",
    author = "Hiroko Kamei",
    note = "dc.publisher: World Scientific Publishing",
    year = "2009",
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    doi = "10.1142/S0218127409025079",
    language = "English",
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    pages = "3707--3732",
    journal = "International Journal of Bifurcation and Chaos in Applied Sciences and Engineering",
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    }

    TY - JOUR

    T1 - The existence and classification of synchrony-breaking bifurcations in regular homogeneous networks using lattice structures

    AU - Kamei, Hiroko

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    N2 - For regular homogeneous networks with simple eigenvalues (real or complex), all possible explicit forms of lattices of balanced equivalence relations can be constructed by introducing lattice generators and lattice indices [Kamei, 2009]. Balanced equivalence relations in the lattice correspond to clusters of partially synchronized cells in a network. In this paper, we restrict attention to regular homogeneous networks with simple real eigenvalues, and one-dimensional internal dynamics for each cell. We first show that lattice elements with nonzero indices indicate the existence of codimension-one synchrony-breaking steady-state bifurcations, and furthermore, the positions of such lattice elements give the number of partially synchronized clusters. Using four-cell regular homogeneous networks as an example, we then classify a large number of regular homogeneous networks into a small number of lattice structures, in which networks share an equivalent clustering type. Indeed, some of these networks even share the same generic bifurcation structure. This classification leads us to explore how regular homogeneous networks that share synchrony-breaking bifurcation structure are topologically related.

    AB - For regular homogeneous networks with simple eigenvalues (real or complex), all possible explicit forms of lattices of balanced equivalence relations can be constructed by introducing lattice generators and lattice indices [Kamei, 2009]. Balanced equivalence relations in the lattice correspond to clusters of partially synchronized cells in a network. In this paper, we restrict attention to regular homogeneous networks with simple real eigenvalues, and one-dimensional internal dynamics for each cell. We first show that lattice elements with nonzero indices indicate the existence of codimension-one synchrony-breaking steady-state bifurcations, and furthermore, the positions of such lattice elements give the number of partially synchronized clusters. Using four-cell regular homogeneous networks as an example, we then classify a large number of regular homogeneous networks into a small number of lattice structures, in which networks share an equivalent clustering type. Indeed, some of these networks even share the same generic bifurcation structure. This classification leads us to explore how regular homogeneous networks that share synchrony-breaking bifurcation structure are topologically related.

    KW - Coupled cell networks

    KW - Lattice

    KW - Synchrony breaking bifurcation

    U2 - 10.1142/S0218127409025079

    DO - 10.1142/S0218127409025079

    M3 - Article

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    JO - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

    JF - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

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