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Coupled cell networks: Semigroups, Lie algebras and normal forms


Authors: Bob Rink and Jan Sanders
Journal: Trans. Amer. Math. Soc. 367 (2015), 3509-3548
MSC (2010): Primary 37G05
DOI: https://doi.org/10.1090/S0002-9947-2014-06221-1
Published electronically: July 21, 2014
MathSciNet review: 3314815
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Abstract: We introduce the concept of a semigroup coupled cell network and show that the collection of semigroup network vector fields forms a Lie algebra. This implies that near a dynamical equilibrium the local normal form of a semigroup network is a semigroup network itself. Networks without the semigroup property will support normal forms with a more general network architecture, but these normal forms nevertheless possess the same symmetries and synchronous solutions as the original network. We explain how to compute Lie brackets and normal forms of coupled cell networks and we characterize the SN-decomposition that determines the normal form symmetry. This paper concludes with a generalization to nonhomogeneous networks with the structure of a semigroupoid.


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Additional Information

Bob Rink
Affiliation: Department of Mathematics, VU University, Amsterdam, The Netherlands
Email: b.w.rink@vu.nl

Jan Sanders
Affiliation: Department of Mathematics, VU University, Amsterdam, The Netherlands
Email: jan.sanders.a@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2014-06221-1
Received by editor(s): September 13, 2012
Received by editor(s) in revised form: June 3, 2013
Published electronically: July 21, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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