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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. 1741. ISSN 14689367
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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 spatiotemporal symmetries of timeperiodic states of groupequivariant 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 spatiotemporal 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 timeperiodic 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 pathconnected 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 pathconnected 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 pathconnected network. We also discuss the nonpathconnected case.
Item Type:  Journal Article  

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:  14689367  
Official Date:  March 2008  
Dates: 


Volume:  Volume 23  
Number:  Number 1  
Number of Pages:  25  
Page Range:  pp. 1741  
Identifier:  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:  DMS0244529 (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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