Coupled cell networks: semigroups, Lie algebras and normal forms

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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.
Original languageEnglish
Pages (from-to)3509-3548
Number of pages40
JournalTransactions of the American Mathematical Society
Volume367
Issue number5
Early online date21 Jul 2014
DOIs
Publication statusPublished - 2014

Bibliographical note

art no PII S0002-9947(2014)06221-1

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