M. Golubitsky, M. Pivato and I. Stewart
Interior symmetry and local bifurcation in coupled cell networks
Dynamical Systems.
19 (4) (2004) 389-407.
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
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 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.
We develop the local bifurcation theory
of coupled cell systems possessing interior
symmetries, by analogy with symmetric bifurcation
theory. The main results are
analogs for `synchrony-breaking' bifurcations
of the Equivariant Branching Lemma for steady-state
bifurcation, and the Equivariant Hopf Theorem for bifurcation
to time-periodic states.