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Circle maps with gaps: Understanding the dynamics of the two-process model for sleep–wake regulation

Published online by Cambridge University Press:  02 May 2018

M. P. BAILEY ,
G. DERKS  and
A. C. SKELDON
M. P. BAILEY
Affiliation:
Department of Mathematics, University of Surrey, Surrey GU2 7XH, UK emails: matthew.bailey@surrey.ac.uk, g.derks@surrey.ac.uk, a.skeldon@surrey.ac.uk
G. DERKS
Affiliation:
Department of Mathematics, University of Surrey, Surrey GU2 7XH, UK emails: matthew.bailey@surrey.ac.uk, g.derks@surrey.ac.uk, a.skeldon@surrey.ac.uk
A. C. SKELDON
Affiliation:
Department of Mathematics, University of Surrey, Surrey GU2 7XH, UK emails: matthew.bailey@surrey.ac.uk, g.derks@surrey.ac.uk, a.skeldon@surrey.ac.uk
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Abstract

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For more than 30 years the ‘two-process model’ has played a central role in the understanding of sleep/wake regulation. This ostensibly simple model is an interesting example of a non-smooth dynamical system, whose rich dynamical structure has been relatively unexplored. The two-process model can be framed as a one-dimensional map of the circle, which, for some parameter regimes, has gaps. We show how border collision bifurcations that arise naturally in maps with gaps extend and supplement the Arnold tongue saddle-node bifurcation set that is a feature of continuous circle maps. The novel picture that results shows how the periodic solutions that are created by saddle-node bifurcations in continuous maps transition to periodic solutions created by period-adding bifurcations as seen in maps with gaps.

Keywords

37E10 Maps of the circle92B25 Biological rhythms and synchronisation37G15 Bifurcations of limit cycles and periodic orbits37E05 Maps of the interval37N25 Dynamical systems in biology
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Special Issue 5: Theory and applications of nonsmooth dynamical systems , October 2018 , pp. 845 - 868
Copyright
Copyright © Cambridge University Press 2018 

Footnotes

†This work was partially supported by the Engineering and Physical Sciences Research Council (Grant number EP/M506655/1).

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