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THE EXISTENCE AND CLASSIFICATION OF SYNCHRONY-BREAKING BIFURCATIONS IN REGULAR HOMOGENEOUS NETWORKS USING LATTICE STRUCTURES

HIROKO KAMEI

MOAC Doctoral Training Centre, University of Warwick, Coventry, CV4 7AL, UK

Current permanent address: Division of Mathematics, University of Dundee, Dundee DD1 4HN.

https://doi.org/10.1142/S0218127409025079Cited by:11 (Source: Crossref)

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.

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