Symmetry and Pattern Formation in Coupled Cell Networks

1999 ◽  
pp. 65-82 ◽  
Author(s):  
Martin Golubitsky ◽  
Ian Stewart
Keyword(s):  
Pattern Formation ◽  
Coupled Cell Networks ◽  
Cell Networks

scholarly journals Coupled cell networks: minimality

PAMM ◽  
2007 ◽  
Vol 7 (1) ◽  
pp. 1030501-1030502
Author(s):  
Manuela A. D. Aguiar ◽  
Ana Paula S. Dias
Keyword(s):  
Coupled Cell Networks ◽  
Cell Networks

Enumerating periodic patterns of synchrony via finite bidirectional networks

2009 ◽  
Vol 466 (2115) ◽  
pp. 891-910
Author(s):  
Ana Paula S. Dias ◽  
Eliana Manuel Pinho
Keyword(s):  
Nearest Neighbour ◽  
Periodic Patterns ◽  
Finite Dimensional ◽  
Spatial Periodicity ◽  
Coupled Cell Networks ◽  
Euclidean Lattice ◽  
Lattice Network ◽  
Bidirectional Networks ◽  
The Given ◽  
Cell Networks

Periodic patterns of synchrony are lattice networks whose cells are coloured according to a local rule, or balanced colouring, and such that the overall system has spatial periodicity. These patterns depict the finite-dimensional flow-invariant subspaces for all the lattice dynamical systems, in the given lattice network, that exhibit those periods. Previous results relate the existence of periodic patterns of synchrony, in n -dimensional Euclidean lattice networks with nearest neighbour coupling architecture, with that of finite coupled cell networks that follow the same colouring rule and have all the couplings bidirectional. This paper addresses the relation between periodic patterns of synchrony and finite bidirectional coloured networks. Given an n -dimensional Euclidean lattice network with nearest neighbour coupling architecture, and a colouring rule with k colours, we enumerate all the periodic patterns of synchrony generated by a given finite network, or graph. This enumeration is constructive and based on the automorphisms group of the graph.

scholarly journals On bifurcations in lifts of regular uniform coupled cell networks

2014 ◽  
Vol 470 (2169) ◽  
pp. 20140241 ◽  
Author(s):  
Célia Sofia Moreira
Keyword(s):  
Bifurcation Point ◽  
Cell System ◽  
Point Of View ◽  
Coupled Cell Networks ◽  
Regular Network ◽  
Cell Networks

A lift of a given network is a network that admits the first network as quotient. Assuming that a bifurcation occurs for a coupled cell system consistent with the structure of a regular network (in which all cells have the same type and receive the same number of inputs and all arrows have the same type), it is well known that some lifts exhibit new bifurcating branches of solutions. In this work, we approach this problem restricting attention to uniform networks, that is, networks that have no loops and no multiple arrows. We show that, from the bifurcation point of view, rings and their lifts are special networks. We also prove that generically there are lifts that just exhibit the bifurcating branches determined by the quotient network and, moreover, we identify all generic situations where lifts exist that may exhibit bifurcating branches that do not appear in the quotient itself.

scholarly journals Patterns of Synchrony in Coupled Cell Networks with Multiple Arrows

2005 ◽  
Vol 4 (1) ◽  
pp. 78-100 ◽  
Author(s):  
Martin Golubitsky ◽  
Ian Stewart ◽  
Andrei Török
Keyword(s):  
Coupled Cell Networks ◽  
Cell Networks

Linear equivalence and ODE-equivalence for coupled cell networks

Nonlinearity ◽  
2005 ◽  
Vol 18 (3) ◽  
pp. 1003-1020 ◽  
Author(s):  
Ana Paula S Dias ◽  
Ian Stewart
Keyword(s):  
Coupled Cell Networks ◽  
Cell Networks

CONSTRUCTION OF LATTICES OF BALANCED EQUIVALENCE RELATIONS FOR REGULAR HOMOGENEOUS NETWORKS USING LATTICE GENERATORS AND LATTICE INDICES

2009 ◽  
Vol 19 (11) ◽  
pp. 3691-3705 ◽  
Author(s):  
HIROKO KAMEI
Keyword(s):  
Special Kind ◽  
Building Blocks ◽  
Equivalence Relations ◽  
Lattice Structures ◽  
Internal Dynamics ◽  
One Dimensional ◽  
Cell Network ◽  
Coupled Cell Networks ◽  
Cell Networks ◽  
Number Of Cells

Regular homogeneous networks are a class of coupled cell network, which comprises one type of cell (node) with one type of coupling (arrow), and each cell has the same number of input arrows (called the valency of the network). In coupled cell networks, robust synchrony (a flow-invariant polydiagonal) corresponds to a special kind of partition of cells, called a balanced equivalence relation. Balanced equivalence relations are determined solely by the network structure. It is well known that the set of balanced equivalence relations on a given finite network forms a complete lattice. In this paper, we consider regular homogeneous networks in which the internal dynamics of each cell is one-dimensional, and whose associated adjacency matrices have simple eigenvalues (real or complex). We construct explicit forms of lattices of balanced equivalence relations for such networks by introducing key building blocks, called lattice generators, along with integer numbers called lattice indices. The properties of lattice indices allow construction of all possible lattice structures for balanced equivalence relations of regular homogeneous networks of any number of cells with any valency. As an illustration, we show all 14 possible lattice structures of balanced equivalence relations for four-cell regular homogeneous networks.

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