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### Periodic dynamics of coupled cell networks II : cyclic symmetry

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Stewart, Ian and Parker, Martyn.
(2008)
*Periodic dynamics of coupled cell networks II : cyclic symmetry.*
Dynamical Systems, Volume 23
(Number 1).
pp. 17-41.
ISSN 1468-9367

**Full text not available from this repository.**

Official URL: http://dx.doi.org/10.1080/14689360701631126

## Abstract

A coupled cell network is a directed graph whose nodes represent dynamical systems and whose directed edges specify how those systems are coupled to each other. The typical dynamic behaviour of a network is strongly constrained by its topology. Especially important constraints arise from global (group) symmetries and local (groupoid) symmetries. The H/K theorem of Buono and Golubitsky characterises the possible spatio-temporal symmetries of time-periodic states of group-equivariant dynamical systems. A version of this theorem for groupsymmetric networks has been proved by Josic and Torok. In networks, spatial symmetries correspond to synchrony of cells, and spatio-temporal symmetries correspond to phase relations between cells. Associated with any coupled cell network is a canonical class of admissible ODEs that respect the network topology. A pattern of synchrony or phase relations in a hyperbolic time-periodic state of such an ODE is rigid if the pattern persists under small admissible perturbations. We characterise rigid patterns of synchrony and rigid phase patterns in coupled cell networks, on the assumption that the periodic state is fully oscillatory (no cell is in equilibrium) and the network has a basic property, the rigid phase property. We conjecture that all networks have the rigid phase property, and that in any path-connected network an admissible ODE with a hyperbolic periodic state can always be perturbed to make the perturbed periodic state fully oscillatory. Our main result states that in any path-connected network with the rigid phase property, every rigid pattern of phase relations can be characterised in two stages. First, sets of cells form synchronous clumps according to a balanced equivalence relation. Second, the corresponding quotient network has a cyclic group of automorphisms, and the phase relations are induced by associating a fixed phase shift with a generator of this group. Thus the clumps of synchronous cells form a discrete rotating wave. As a corollary, we prove an analogue of the H/K theorem for any path-connected network. We also discuss the non-path-connected case.

Item Type: | Journal Article |
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Subjects: | Q Science > QA Mathematics T Technology > TJ Mechanical engineering and machinery |

Divisions: | Faculty of Science > Mathematics |

Library of Congress Subject Headings (LCSH): | Directed graphs, Differentiable dynamical systems, System analysis |

Journal or Publication Title: | Dynamical Systems |

Publisher: | Taylor & Francis Ltd. |

ISSN: | 1468-9367 |

Official Date: | March 2008 |

Volume: | Volume 23 |

Number: | Number 1 |

Number of Pages: | 25 |

Page Range: | pp. 17-41 |

Identification Number: | 10.1080/14689360701631126 |

Status: | Peer Reviewed |

Publication Status: | Published |

Access rights to Published version: | Restricted or Subscription Access |

Funder: | National Science Foundation (U.S.) (NSF), Engineering and Physical Sciences Research Council (EPSRC) |

Grant number: | DMS-0244529 (NSF) |

References: | [1] Antoneli, F., Dias, A.P.S., Golubitsky, M. and Wang, Y., 2005, Flow invariant subspaces |

URI: | http://wrap.warwick.ac.uk/id/eprint/30081 |

Data sourced from Thomson Reuters' Web of Knowledge

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