2013/01/31 by Leonhard Lücken, Jan Philipp Pade, Kolja Knauer · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Applied mathematics #Attractor #Class (philosophy) #Combinatorics #Computer science #Delay differential equation #Differential equation #Dimension (graph theory) #Discrete mathematics #Dynamical system (definition) #Dynamical systems theory #Equivalence (formal languages) #Equivalence class (music) #Gene Regulatory Network Analysis #Graph #Invariant (physics) #Mathematical analysis #Mathematics #Neural Networks Stability and Synchronization #Nonlinear Dynamics and Pattern Formation #Pure mathematics #Topology (electrical circuits) #Transformation (genetics) #math.DS #msc:34K17 #msc:34K20 #msc:37L05 #msc:37L15
paper · pdf · doi:10.1137/14097183x
published as SIAM Journal on Applied Dynamical Systems 2015 14:1, 286-304
openalex publication_date 2015/01/01 · arxiv created 2015/04/16 · arxiv updated 2016/02/01 · openalex created_date 2019/06/27 · openalex updated_date 2026/08/05
In this article we study networks of coupled dynamical systems with time-delayed connections. If two such networks hold different delays on the connections, it is in general possible that they exhibit different dynamical behavior as well. We prove that for particular sets of delays this is not the case. To this aim we introduce a componentwise timeshift transformation (CTT) which allows us to classify systems which possess equivalent dynamics, though possibly different sets of connection delays. In particular, we show for a large class of semiflows (including the case of delay differential equations) that the stability of attractors is invariant under this transformation. Moreover we show that each equivalence class which is mediated by the CTT possesses a representative system in which the number of different delays is not larger than the cycle space dimension of the underlying graph. We conclude that the “true” dimension of the corresponding parameter space of delays is in general smaller than it appears at first glance.