Approximating the Steady-State Periodic Solutions of Contractive Systems

Samuel Coogan, Michael Margaliot*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We consider contractive systems whose trajectories evolve on a compact and convex state-space. It is well-known that if the time-varying vector field of the system is periodic, then the system admits a unique globally asymptotically stable periodic solution. Obtaining explicit information on this periodic solution and its dependence on various parameters is important both theoretically and in numerous applications. We develop an approach for approximating such a periodic trajectory using the periodic trajectory of a simpler system (e.g., an LTI system). The approximation includes an error bound that is based on the input-to-state stability property of contractive systems. We show that in some cases, this error bound can be computed explicitly. We also use the bound to derive a new theoretical result, namely, that a contractive system with an additive periodic input behaves like a low-pass filter. We demonstrate our results using several examples from systems biology.

Original languageEnglish
Article number8360452
Pages (from-to)847-853
Number of pages7
JournalIEEE Transactions on Automatic Control
Issue number2
StatePublished - Feb 2019


  • Computational methods
  • entrainment
  • stability of nonlinear systems
  • systems biology


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