2012/09/14 by Bob Rink, Rink, Bob, Jan A. Sanders +2 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Physics and Astronomy · #Gene Regulatory Network Analysis #Neural Networks Stability and Synchronization #Nonlinear Dynamics and Pattern Formation #math.DS #msc:20M30 #msc:34D06 #msc:37G05 #msc:37G40 #nlin.CD
paper · pdf · doi:10.48550/arxiv.1209.3209
arxiv created 2012/09/14 · arxiv updated 2012/09/17
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.