Contraction after small transients

Michael Margaliot, Eduardo D. Sontag, Tamir Tuller

Research output: Contribution to journalArticlepeer-review


Contraction theory is a powerful tool for proving asymptotic properties of nonlinear dynamical systems including convergence to an attractor and entrainment to a periodic excitation. We consider three generalizations of contraction with respect to a norm that allow contraction to take place after small transients in time and/or amplitude. These generalized contractive systems (GCSs) are useful for several reasons. First, we show that there exist simple and checkable conditions guaranteeing that a system is a GCS, and demonstrate their usefulness using several models from systems biology. Second, allowing small transients does not destroy the important asymptotic properties of contractive systems like convergence to a unique equilibrium point, if it exists, and entrainment to a periodic excitation. Third, in some cases as we change the parameters in a contractive system it becomes a GCS just before it looses contractivity with respect to a norm. In this respect, generalized contractivity is the analogue of marginal stability in Lyapunov stability theory.

Original languageEnglish
Pages (from-to)178-184
Number of pages7
StatePublished - May 2016


  • Contraction
  • Differential analysis
  • Entrainment
  • Phase locking
  • Stability
  • Systems biology


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