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Zeno breaking, the Contact effect and sensitive behaviour in piecewise-linear systems

Published online by Cambridge University Press:  21 March 2018

R. EDWARDS
R. EDWARDS*
Affiliation:
Department of Mathematics, University of Victoria, PO Box 1700 STN CSC, Victoria, BC V8W 2Y2, Canada email: edwards@uvic.ca
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Abstract

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Non-smooth approximations of steep sigmoidal switching networks, such as those used as qualitative models of gene regulation, lead to analytic and computational challenges that arise as a result of the discontinuities in the vector fields. In order to highlight the need for care in dealing with such systems, several particular phenomena are presented here through illustrative examples, including ‘Zeno breaking’, or computing beyond the finite time convergence of an infinite sequence of threshold transitions; the ‘Contact’ effect, in which in the discontinuous limit, trajectories can pass through a ‘saddle point’ without stopping, though these solutions are not unique and other solutions stop for arbitrary time intervals; and sensitive behaviour that arises from exotic dynamics within switching regions.

Keywords

Discontinuous equationssingular perturbationsdifferential inclusionsoscillationsdynamical systems in biology (34A36; 34D15; 34A60; 34C10; 37N25)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Special Issue 5: Theory and applications of nonsmooth dynamical systems , October 2018 , pp. 826 - 844
Copyright
Copyright © Cambridge University Press 2018 

Footnotes

†This work was partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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