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

Bailey, Matthew, Derks, Gerardina and Skeldon, Anne (2018) Circle maps with gaps: understanding the dynamics of the two process model for sleep-wake regulation European Journal of Applied Mathematics.

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Abstract

For more than thirty years the `two process model' has played a central role in the under- standing of sleep/wake regulation. This ostensibly simple model is an interesting example of a nonsmooth dynamical system whose rich dynamical structure has been relatively un- explored. 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.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
NameEmailORCID
Bailey, Matthewmatthew.bailey@surrey.ac.uk
Derks, GerardinaG.Derks@surrey.ac.uk
Skeldon, AnneA.Skeldon@surrey.ac.uk
Date : 2 May 2018
Funders : Engineering and Physical Sciences Research Council (EPSRC)
Identification Number : 10.1017/S0956792518000190
Copyright Disclaimer : Copyright 2018 Cambridge University Press
Uncontrolled Keywords : 37E10 Maps of the circle; 92B25 Biological rhythms and synchronisation; 37G15 Bifurcations of limit cycles and periodic orbits; 37E05 Maps of the interval; 37N25 Dynamical systems in biology.
Depositing User : Clive Harris
Date Deposited : 01 Mar 2018 19:36
Last Modified : 09 May 2018 10:30
URI: http://epubs.surrey.ac.uk/id/eprint/845934

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