Bifurcation analysis of delay-induced periodic oscillations

In this paper we consider a generic differential equation with a cubic nonlinearity and delay. This system, in the absence of delay, is known to undergo an oscillatory instability. The addition of the delay is shown to result in the creation of a number of periodic solutions with constant amplitude...

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Bibliographic Details
Published in:Journal of computational and applied mathematics Vol. 233; no. 9; pp. 2405 - 2412
Main Author: Green, K.
Format: Journal Article
Language:English
Published: Kidlington Elsevier B.V 01.03.2010
Elsevier
Subjects:
Codimension-one and codimension-two bifurcations
Delay differential equations
Exact sciences and technology
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis in abstract spaces
Numerical analysis. Scientific computation
Ordinary differential equations
Organising centres
Sciences and techniques of general use
Delay differential equations
Codimension-one and codimension-two bifurcations
Organising centres
Differential equation
Bifurcation
Delay system
Periodic solution
Delay equation
Numerical analysis
Feedback
Applied mathematics
Oscillation
Nonlinearity
Abstract space
Mathematical model
ISSN:0377-0427, 1879-1778
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Summary:In this paper we consider a generic differential equation with a cubic nonlinearity and delay. This system, in the absence of delay, is known to undergo an oscillatory instability. The addition of the delay is shown to result in the creation of a number of periodic solutions with constant amplitude and a constant frequency; the number of solutions increases with the size of the delay. Indeed, for many physical applications in which oscillatory instabilities are induced by a delayed response or feedback mechanism, the system under consideration forms the underlying backbone for a mathematical model. Our study showcases the effectiveness of performing a numerical bifurcation analysis, alongside the use of analytical and geometrical arguments, in investigating systems with delay. We identify curves of codimension-one bifurcations of periodic solutions. We show how these curves interact via codimension-two bifurcation points: double singularities which organise the bifurcations and dynamics in their local vicinity.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2009.10.025