Global dynamics for switching systems and their extensions by linear differential equations

Zane Huttinga, Bree Cummins, Tomáš Gedeon, Konstantin Mischaikow

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


Switching systems use piecewise constant nonlinearities to model gene regulatory networks. This choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of Morse graphs associated to nodes of a parameter graph. The parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical Morse graphs. However, there are many cellular processes that do not exhibit threshold-like behavior and thus are not well described by a switching system. We consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations. We show that the parameter graphs associated to the switching system and any of its extensions are identical. For each parameter graph node, there is an order-preserving map from the Morse graph of the switching system to the Morse graph of any of its extensions. We provide counterexamples that show why possible stronger relationships between the Morse graphs are not valid.

Original languageEnglish (US)
Pages (from-to)19-37
Number of pages19
JournalPhysica D: Nonlinear Phenomena
StatePublished - Mar 15 2018

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


  • Gene regulation
  • Morse graphs
  • Switching systems
  • Transcription/translation model


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