Interior symmetry and local bifurcation in coupled cell networks

Abstract

A coupled cell system is a network of dynamical systems, or 'cells', coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells and edges that preserves all internal dynamics and all couplings. It is well known that symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. Recently, the introduction of a less stringent form of symmetry, the 'symmetry groupoid', has shown that global group-theoretic symmetry is not the only mechanism that can create such states in a coupled cell system. The symmetry groupoid consists of structure-preserving bijections between certain subsets of the cell network, the input sets. Here, we introduce a concept intermediate between the groupoid symmetries and the global group symmetries of a network: 'interior symmetry'. This concept is closely related to the groupoid structure, but imposes stronger constraints of a group-theoretic nature. We develop the local bifurcation theory of coupled cell systems possessing interior symmetries, by analogy with symmetric bifurcation theory. The main results are analogues for 'synchrony-breaking' bifurcations of the Equivariant Branching Lemma for steady-state bifurcation, and the Equivariant Hopf Theorem for bifurcation to time-periodic states.

Quasi-periodic states in coupled rings of cells

“…The notion of interior symmetry, recently introduced by Golubitsky et al [14], generalizes the usual definition of symmetry. The schematic diagram on the right of Fig.…”

Nonlinear dynamics of networks: the groupoid formalism

“…It turns out that there is a useful intermediate concept, a strengthening of input-equivalence 'symmetries' that we call interior symmetry. It was introduced in [33], and in particular it leads to some systematic results on local bifurcation. Like all local bifurcation theories, a key role is played by the eigenvalue structure of linear (admissible) vector fields, and this section motivates the one that follows it on the linear theory.…”

“…We manage to bring a semblance of order into one tiny corner of this vast area in Section 12, which introduces a type of network symmetry that is somehow intermediate between global group symmetry and local groupoid symmetry. We call it 'interior symmetry', and we discuss analogues of the Equivariant Branching Lemma for steady-state bifurcation and the Equivariant Hopf Theorem for bifurcation to periodic states proved in [33].…”

Heteroclinic Networks in Homogeneous and Heterogeneous Identical Cell Systems

“…A general theory of networks of coupled cells has been formalized by Stewart et al (2003), Golubitsky et al (2004), Golubitsky et al (2005) (a survey and review of some of this work appears in Golubitsky and Stewart (2006)). Their approach is relatively algebraic in character and depends on groupoid formalism, graphs and the idea of a quotient network.…”