### Abstract

A coupled cell network describes interacting (coupled) individual systems (cells). As in networks from real applications, coupled cell networks can represent inhomogeneous networks where different types of cells interact with each other in different ways, which can be represented graphically by different symbols, or abstractly by equivalence relations. Various synchronous behaviors, from full synchrony to partial synchrony, can be observed for a given network. Patterns of synchrony, which do not depend on specific dynamics of the network, but only on the network structure, are associated with a special type of partition of cells, termed balanced equivalence relations. Algorithms in Aldis [J. W. Aldis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), pp. 407-427] and Belykh and Hasler [I. Belykh and M. Hasler, Chaos, 21 (2011), 016106] find the unique pattern of synchrony with the fewest clusters. In this paper, we compute the set of all possible patterns of synchrony and show their hierarchy structure as a complete lattice. We represent the network structure of a given coupled cell network by a symbolic adjacency matrix encoding the different coupling types. We show that balanced equivalence relations can be determined by a matrix computation on the adjacency matrix which forms a block structure for each balanced equivalence relation. This leads to a computer algorithm to search for all possible balanced equivalence relations. Our computer program outputs the balanced equivalence relations, quotient matrices, and a complete lattice for user specified coupled cell networks. Finding the balanced equivalence relations of any network of up to 15 nodes is tractable, but for larger networks this depends on the pattern of synchrony with the fewest clusters.

Original language | English |
---|---|

Pages (from-to) | 352-382 |

Number of pages | 31 |

Journal | Siam Journal on Applied Dynamical Systems |

Volume | 12 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 |

### Keywords

- SYSTEMS
- SYNCHRONIZATION
- lattice
- coupled cell networks
- OSCILLATORS
- COMPLEX NETWORKS
- synchrony
- PATTERNS
- BIFURCATIONS
- DYNAMICS
- balanced equivalence relations

### Cite this

*Siam Journal on Applied Dynamical Systems*,

*12*(1), 352-382. https://doi.org/10.1137/100819795

}

*Siam Journal on Applied Dynamical Systems*, vol. 12, no. 1, pp. 352-382. https://doi.org/10.1137/100819795

**Computation of balanced equivalence relations and their lattice for a coupled cell network.** / Kamei, Hiroko; Cock, Peter J. A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Computation of balanced equivalence relations and their lattice for a coupled cell network

AU - Kamei, Hiroko

AU - Cock, Peter J. A.

PY - 2013

Y1 - 2013

N2 - A coupled cell network describes interacting (coupled) individual systems (cells). As in networks from real applications, coupled cell networks can represent inhomogeneous networks where different types of cells interact with each other in different ways, which can be represented graphically by different symbols, or abstractly by equivalence relations. Various synchronous behaviors, from full synchrony to partial synchrony, can be observed for a given network. Patterns of synchrony, which do not depend on specific dynamics of the network, but only on the network structure, are associated with a special type of partition of cells, termed balanced equivalence relations. Algorithms in Aldis [J. W. Aldis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), pp. 407-427] and Belykh and Hasler [I. Belykh and M. Hasler, Chaos, 21 (2011), 016106] find the unique pattern of synchrony with the fewest clusters. In this paper, we compute the set of all possible patterns of synchrony and show their hierarchy structure as a complete lattice. We represent the network structure of a given coupled cell network by a symbolic adjacency matrix encoding the different coupling types. We show that balanced equivalence relations can be determined by a matrix computation on the adjacency matrix which forms a block structure for each balanced equivalence relation. This leads to a computer algorithm to search for all possible balanced equivalence relations. Our computer program outputs the balanced equivalence relations, quotient matrices, and a complete lattice for user specified coupled cell networks. Finding the balanced equivalence relations of any network of up to 15 nodes is tractable, but for larger networks this depends on the pattern of synchrony with the fewest clusters.

AB - A coupled cell network describes interacting (coupled) individual systems (cells). As in networks from real applications, coupled cell networks can represent inhomogeneous networks where different types of cells interact with each other in different ways, which can be represented graphically by different symbols, or abstractly by equivalence relations. Various synchronous behaviors, from full synchrony to partial synchrony, can be observed for a given network. Patterns of synchrony, which do not depend on specific dynamics of the network, but only on the network structure, are associated with a special type of partition of cells, termed balanced equivalence relations. Algorithms in Aldis [J. W. Aldis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), pp. 407-427] and Belykh and Hasler [I. Belykh and M. Hasler, Chaos, 21 (2011), 016106] find the unique pattern of synchrony with the fewest clusters. In this paper, we compute the set of all possible patterns of synchrony and show their hierarchy structure as a complete lattice. We represent the network structure of a given coupled cell network by a symbolic adjacency matrix encoding the different coupling types. We show that balanced equivalence relations can be determined by a matrix computation on the adjacency matrix which forms a block structure for each balanced equivalence relation. This leads to a computer algorithm to search for all possible balanced equivalence relations. Our computer program outputs the balanced equivalence relations, quotient matrices, and a complete lattice for user specified coupled cell networks. Finding the balanced equivalence relations of any network of up to 15 nodes is tractable, but for larger networks this depends on the pattern of synchrony with the fewest clusters.

KW - SYSTEMS

KW - SYNCHRONIZATION

KW - lattice

KW - coupled cell networks

KW - OSCILLATORS

KW - COMPLEX NETWORKS

KW - synchrony

KW - PATTERNS

KW - BIFURCATIONS

KW - DYNAMICS

KW - balanced equivalence relations

U2 - 10.1137/100819795

DO - 10.1137/100819795

M3 - Article

VL - 12

SP - 352

EP - 382

JO - Siam Journal on Applied Dynamical Systems

JF - Siam Journal on Applied Dynamical Systems

SN - 1536-0040

IS - 1

ER -