Abstract
In this paper, we study degenerate Hopf bifurcations in a class of parametrized retarded functional differential equations. Specifically, we are interested in the case where the eigenvalue crossing condition of the classical Hopf bifurcation theorem is violated. Our approach is based on center manifold reduction and Poincaré–Birkhoff normal forms, and a singularity theoretical classification of this degenerate Hopf bifurcation. Our results are applied to a recently developed SIS model incorporating a delayed behavioral response. We show the phenomenon of endemic bubbles, which is characterized by a branch of periodic solutions which bifurcates from the endemic equilibrium at some value of the basic reproduction number \(R_0\), and then reconnects to the endemic equilibrium at a larger value of \(R_0\), originates in a codimension-two organizing center where the eigenvalue crossing condition for the Hopf bifurcation theorem is violated.
Similar content being viewed by others
References
Beuter, A., Bélair, J., Labrie, C.: Feedback and delays in neurological diseases: a modeling study using dynamical systems. Bull. Math. Biol. 55, 525–541 (1993)
Bi, P., Ruan, S.: Bifurcations in delay differential equations and applications to tumor and immune system interaction models. SIAM J. Appl. Dyn. Syst. 12, 1847–1888 (2013)
Brown, G., Postlethwaite, C.M., Silber, M.: Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation. Phys. D 240, 859–871 (2011)
Faria, T., Magalhães, L.T.: Normal forms for retarded functional differential equations and applications to Bogdanov–Takens singularity. J. Differ. Equ. 122, 201–224 (1995)
Faria, T., Magalhães, L.T.: Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation. J. Differ. Equ. 122, 181–200 (1995)
Golubitsky, M., Langford, W.F.: Classification and unfoldings of degenerate Hopf bifurcations. J. Differ. Equ. 41, 375–415 (1981)
Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, Vol. II. Applied Mathematical Sciences, vol. 69. Springer, New York (1988)
Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Applied Mathematical Sciences, vol. 99. Springer, New York (1993)
Heil, T., Fischer, I., Elsäßer, W., Krauskopf, B., Green, K., Gavrielides, A.: Delay dynamics of semiconductor lasers with short external cavities: bifurcation scenarios and mechanisms. Phys. Rev. E 67, 066214-1–066214-11 (2003)
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Mathematics in Science and Engineering, vol. 191. Academic Press, Boston (1993)
Liu, M., Liz, E., Röst, G.: Endemic bubbles generated by delayed behavioral response: global stability and bifurcation switches in an SIS model. SIAM J. Appl. Math. 75, 75–91 (2015)
Longtin, A., Milton, J.G.: Modeling autonomous oscillations in the human pupil light reflex using nonlinear delay-differential equations. Bull. Math. Biol. 51, 605–624 (1989)
Ma, J., Tu, H.: Analysis of the stability and Hopf bifurcation of money supply delay in complex macroeconomics models. Nonlinear Dyn. 76, 497–508 (2014)
Marsden, J.E., McCracken, M.F.: The Hopf Bifurcation and Its Applications. Springer, New York (1976)
Neamtu, M., Opris, D., Chilarescu, C.: Hopf bifurcation in a dynamic IS-LM model with time delay. Chaos Solitons Fractals 34, 519–530 (2007)
Purewal, A.S., Postlethwaite, C.M., Krauskopf, B.: A global bifurcation analysis of the subcritical Hopf normal form subject to Pyragas time-delayed feedback control. SIAM J. Appl. Dyn. Syst. 13, 1879–1915 (2014)
Qesmi, R., Babram, M.Ait, Hbid, M.L.: A Maple program for computing a terms of a center manifold, and element of bifurcations for a class of retarded functional differential equations with Hopf singularity. Appl. Math. Comput. 175, 932–968 (2006)
Sieber, J., Krauskopf, B.: Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity. Nonlinearity 17, 85–103 (2004)
Stone, E., Campbell, S.A.: Stability and bifurcation analysis of a nonlinear DDE model for drilling. J. Nonlinear Sci. 14, 27–57 (2004)
Suarez, M.J., Schopf, P.L.: A delayed action oscillator for ENSO. J. Atmos. Sci. 45, 3283–3287 (1988)
Vladimirov, A.G., Turaev, D., Kozyreff, G.: Delay differential equations for mode-locked semiconductor lasers. Opt. Lett. 29, 1221–1223 (2004)
Acknowledgments
This research is partly supported by the Natural Sciences and Engineering Research Council of Canada in the form of a Discovery Grant. The author is grateful to the reviewers for carefully reading the manuscript and for providing valuable suggestions that have considerably improved the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Sue Ann Campbell.
Appendix: First Lyapunov Coefficient
Appendix: First Lyapunov Coefficient
For the delay-differential equation (2.18), formulae (2.15), (2.16) and (2.19) yield the Lyapunov coefficient \(K_1\) in terms of the coefficients \(f_{(j,k)}\) in (2.19) and \(\alpha ^*\), \(\beta ^*\), \(\omega ^*\) and \(\tau \). The following result was computed using the symbolic computation software package Maple:
It follows that in order for \(K_1\) to be well defined, we need the non-degeneracy conditions \(\alpha ^*+\beta ^*\ne 0\) and \(4\alpha ^*-5\beta ^*\ne 0\). For a generic delay-differential equation of the form (2.18), this coefficient \(K_1\) will be nonzero.
Rights and permissions
About this article
Cite this article
LeBlanc, V.G. A Degenerate Hopf Bifurcation in Retarded Functional Differential Equations, and Applications to Endemic Bubbles. J Nonlinear Sci 26, 1–25 (2016). https://doi.org/10.1007/s00332-015-9266-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-015-9266-5