Abstract
Periodic motions in DDE (Differential-Delay Equations) are typically created in Hopf bifurcations. In this chapter we examine this process from several points of view. Firstly we use Lindstedt’s perturbation method to derive the Hopf Bifurcation Formula, which determines the stability of the periodic motion. Then we use the Two Variable Expansion Method (also known as Multiple Scales) to investigate the transient behavior involved in the approach to the periodic motion. Next we use Center Manifold Analysis to reduce the DDE from an infinite dimensional evolution equation on a function space to a two dimensional ODE (Ordinary Differential Equation) on the center manifold, the latter being a surface tangent to the eigenspace associated with the Hopf bifurcation. Finally we provide an application to gene copying in which the delay is due to an observed time lag in the transcription process.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Camacho E., Rand R. and Howland, H., 2004, Dynamics of two van der Pol oscillators coupled via a bath, International J. Solids and Structures, 41, 2133–2143.
Campbell S.A., Belair J., Ohira T. and Milton J., 1995, Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback, Chaos, 5, 640–645.
Casal A. and Freedman M., 1980, A Poincare-Lindstedt approach to bifurcation problems for differential-delay equations, IEEE Transactions on Automatic Control, 25, 967–973.
Cole J.D., 1968, Perturbation Methods in Applied Mathematics, Blaisdell, Waltham, MA.
Das S.L. and Chatterjee A., 2002, Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations, Nonlinear Dynamics, 30, 323–335.
Das S.L. and Chatterjee A., 2005, Second order multiple scales for oscillators with large delay, Nonlinear Dynamics, 39, 375–394.
Guckenheimer J. and Holmes P., 1983, Nonlinear Oscillations, Dynamical Systems; and Bifurcations of Vector Fields, Springer, New York.
Hassard B.D., Kazarinoff N.D. and Wan Y-H., 1981, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge.
Kalmar-Nagy T., Stepan G. and Moon F.C., 2001, Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations, Nonlinear Dynamics, 26, 121–142.
Kot M., 1979, Delay-differential Models and the Effects of Economic Delays on the Stability of Open-access Fisheries, M.S. thesis, Cornell University
Mahaffy J.M., 1988, Genetic control models with diffusion and delays, Mathematical Biosciences, 90, 519–533.
Mahaffy J.M., Jorgensen D.A. and Vanderheyden R.L., 1992, Oscillations in a model of repression with external control, J. Math. Biology, 30, 669–691.
Mocek W.T., Rudbicki R. and Voit E.O., 2005, Approximation of delays in biochemical systems, Mathematical Biosciences, 198, 190–216.
Monk N.A.M., 2003, Oscillatory expreSSIon of Hes1, p53, and NF-KB driven by transcriptional time delays, Current Biology, 13, 1409–1413.
Nayfeh A.H., 1973, Perturbation Methods, Wiley, New York.
Nayfeh A.H., 2008, Order reduction of retarded nonlinear systems — the method of multiple scales versus center-manifold reduction, Nonlinear Dynamics, 51, 483–500.
Rand R.H. and Armbruster D., 1987, Perturbation Methods, Bifurcation Theory and Computer Algebra, Springer, New York.
Rand R.H., 2005, Lecture Notes on Nonlinear Vibrations (version 52), available online at http://audiophile.tam.comell.edu/randdocs.
Rand R. and Verdugo A., 2007, Hopf Bifurcation formula for first order differentialdelay equations, Communications in Nonlinear Science and Numerical Simulation, 12, 859–864.
Sanders J.A. and Verhulst F., 1985, Averaging Methods in Nonlinear Dynamical Systems, Springer, New York.
Stepan G., 1989, Retarded Dynamical Systems: Stability and Characteristic Functions, Longman Scientific and Technical, Essex.
Verdugo A. and Rand R., 2008a, Hopf bifurcation in a DDE model of gene expression, Communications in Nonlinear Science and Numerical Simulation, 13, 235–242.
Verdugo A. and Rand R., 2008b, Center manifold analysis of a DDE model of gene expression, Communications in Nonlinear Science and Numerical Simulation, 13, 1112–11120.
Wang H. and Hu H., 2003, Remarks on the perturbation methods in solving the second-order delay differential equations, Nonlinear Dynamics, 33, 379–398.
Wirkus S. and Rand R., 2002, Dynamics of two coupled van der Pol oscillators with delay coupling, Nonlinear Dynamics, 30, 205–221.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Rand, R. (2011). Differential-Delay Equations. In: Luo, A.C.J., Sun, JQ. (eds) Complex Systems. Nonlinear Physical Science, vol 0. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17593-0_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-17593-0_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17592-3
Online ISBN: 978-3-642-17593-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)