112 research outputs found
Symmetry groupoids and patterns of synchrony in coupled cell networks
A coupled cell system is a network of dynamical systems, or âcells,â 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. Symmetry can lead to patterns of synchronized
cells, rotating waves, multirhythms, and synchronized chaos. We ask whether symmetry is the only
mechanism that can create such states in a coupled cell system and show that it is not.
The key idea is to replace the symmetry group by the symmetry groupoid, which encodes information
about the input sets of cells. (The input set of a cell consists of that cell and all cells
connected to that cell.) The admissible vector fields for a given graphâthe dynamical systems with
the corresponding internal dynamics and couplingsâare precisely those that are equivariant under
the symmetry groupoid. A pattern of synchrony is ârobustâ if it arises for all admissible vector
fields. The first main result shows that robust patterns of synchrony (invariance of âpolydiagonalâ
subspaces under all admissible vector fields) are equivalent to the combinatorial condition that an
equivalence relation on cells is âbalanced.â The second main result shows that admissible vector
fields restricted to polydiagonal subspaces are themselves admissible vector fields for a new coupled
cell network, the âquotient network.â The existence of quotient networks has surprising implications
for synchronous dynamics in coupled cell systems
Symmetry groupoids and admissible vector fields for coupled cell networks
The space of admissible vector fields, consistent with the structure of a network of coupled dynamical systems, can be specified in terms of the network's symmetry groupoid. The symmetry groupoid also determines the robust patterns of synchrony in the network â those that arise because of the network topology. In particular, synchronous cells can be identified in a canonical manner to yield a quotient network. Admissible vector fields on the original network induce admissible vector fields on the quotient, and any dynamical state of such an induced vector field can be lifted to the original network, yielding an analogous state in which certain sets of cells are synchronized. In the paper, necessary and sufficient conditions are specified for all admissible vector fields on the quotient to lift in this manner. These conditions are combinatorial in nature, and the proof uses invariant theory for the symmetric group. Also the symmetry groupoid of a quotient is related to that of the original network, and it is shown that there is a close analogy with the usual normalizer symmetry that arises in group-equivariant dynamics
Amplified Hopf bifurcations in feed-forward networks
In a previous paper, the authors developed a method for computing normal
forms of dynamical systems with a coupled cell network structure. We now apply
this theory to one-parameter families of homogeneous feed-forward chains with
2-dimensional cells. Our main result is that Hopf bifurcations in such families
generically generate branches of periodic solutions with amplitudes growing
like , , , etc. Such amplified
Hopf branches were previously found by others in a subclass of feed-forward
networks with three cells, first under a normal form assumption and later by
explicit computations. We explain here how these bifurcations arise generically
in a broader class of feed-forward chains of arbitrary length
The lattice of balanced equivalence relations of a coupled cell network
A coupled cell system is a collection of dynamical systems, or âcellsâ, that are coupled together. The associated coupled cell network is a labelled directed graph that indicates how the cells are coupled, and which cells are equivalent. Golubitsky, Stewart, Pivato and TĂśrĂśk have presented a framework for coupled cell systems that permits a classification of robust synchrony in terms of the concept of a âbalanced equivalence relationâ, which depends solely on the network architecture. In their approach the network is assumed to be finite. We prove that the set of all balanced equivalence relations on a network forms a lattice, in the sense of a partially ordered set in which any two elements have a meet and a join. The partial order is defined by refinement. Some aspects of the theory make use of infinite networks, so we work in the category of networks of âfinite typeâ, a class that includes all locally finite networks. This context requires some modifications to the standard framework. As partial compensation, the lattice of balanced equivalence relations can then be proved complete. However, the intersection of two balanced equivalence relations need not be balanced, as we show by a simple example, so this lattice is not a sublattice of the lattice of all equivalence relations with its usual operations of meet and join. We discuss the structure of this lattice and computational issues associated with it. In particular, we describe how to determine whether the lattice contains more than the equality relation. As an example, we derive the form of the lattice for a linear chain of identical cells with feedback
Dynamics of coupled cell networks: synchrony, heteroclinic cycles and inflation
Copyright Š 2011 Springer. The final publication is available at www.springerlink.comWe consider the dynamics of small networks of coupled cells. We usually assume asymmetric inputs and no global or local symmetries in the network and consider equivalence of networks in this setting; that is, when two networks with different architectures give rise to the same set of possible dynamics. Focussing on transitive (strongly connected) networks that have only one type of cell (identical cell networks) we address three questions relating the network structure to dynamics. The first question is how the structure of the network may force the existence of invariant subspaces (synchrony subspaces). The second question is how these invariant subspaces can support robust heteroclinic attractors. Finally, we investigate how the dynamics of coupled cell networks with different structures and numbers of cells can be related; in particular we consider the sets of possible âinflationsâ of a coupled cell network that are obtained by replacing one cell by many of the same type, in such a way that the original network dynamics is still present within a synchrony subspace. We illustrate the results with a number of examples of networks of up to six cells
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