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Periodic solutions and their bifurcations in a non-smooth second-order delay differential equation

Periodic solutions and their bifurcations in a non-smooth second-order delay differential equation
Periodic solutions and their bifurcations in a non-smooth second-order delay differential equation
We consider a non-smooth second order delay differential equation (DDE) that was previously studied as a model of the pupil light reflex. It can also be viewed as a prototype model for a system operated under delayed relay control. We use the explicit construction of solutions of the non-smooth DDE hand-in-hand with a numerical continuation study of a related smoothed system. This allows us to produce a comprehensive global picture of the dynamics and bifurcations, which extends and completes previous results. Specifically, we find a rich combinatorial structure consisting of solution branches connected at resonance points. All new solutions of the smoothed system were subsequently constructed as solutions of the non-smooth system. Furthermore, we show an example of the unfolding in the smoothed system of a non-smooth bifurcation point, from which infinitely many solution branches emanate. This shows that smoothing of the DDE may provide insight even into bifurcations that can only occur in non-smooth systems
1468-9367
289-311
Barton, DAW
4b58c886-bdcf-42bc-b880-ddab7d136081
Krauskopf, B
2a89c58d-4f47-4a21-94f1-6245b1a62c2e
Wilson, R. Eddie
01d7f1f2-f8ee-4661-b713-dcefcddb89bd
Barton, DAW
4b58c886-bdcf-42bc-b880-ddab7d136081
Krauskopf, B
2a89c58d-4f47-4a21-94f1-6245b1a62c2e
Wilson, R. Eddie
01d7f1f2-f8ee-4661-b713-dcefcddb89bd

Barton, DAW, Krauskopf, B and Wilson, R. Eddie (2006) Periodic solutions and their bifurcations in a non-smooth second-order delay differential equation. Dynamical Systems, 21 (3), 289-311. (doi:10.1080/14689360500539363).

Record type: Article

Abstract

We consider a non-smooth second order delay differential equation (DDE) that was previously studied as a model of the pupil light reflex. It can also be viewed as a prototype model for a system operated under delayed relay control. We use the explicit construction of solutions of the non-smooth DDE hand-in-hand with a numerical continuation study of a related smoothed system. This allows us to produce a comprehensive global picture of the dynamics and bifurcations, which extends and completes previous results. Specifically, we find a rich combinatorial structure consisting of solution branches connected at resonance points. All new solutions of the smoothed system were subsequently constructed as solutions of the non-smooth system. Furthermore, we show an example of the unfolding in the smoothed system of a non-smooth bifurcation point, from which infinitely many solution branches emanate. This shows that smoothing of the DDE may provide insight even into bifurcations that can only occur in non-smooth systems

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Published date: 2006

Identifiers

Local EPrints ID: 184259
URI: http://eprints.soton.ac.uk/id/eprint/184259
ISSN: 1468-9367
PURE UUID: 323af852-1e3b-4fa3-bf48-fc673a1b988a

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Date deposited: 12 May 2011 11:47
Last modified: 16 Jul 2019 23:40

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