Journal
AUTOMATICA
Volume 67, Issue -, Pages 178-184Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.automatica.2016.01.018
Keywords
Differential analysis; Contraction; Stability; Entrainment; Phase locking; Systems biology
Funding
- Israeli Ministry of Science, Technology and Space
- [NIH 1R01GM100473]
- [AFOSR FA9550-14-1-0060]
- [ONR N00014-13-1-0074]
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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. (C) 2016 Elsevier Ltd. All rights reserved.
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