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A Degenerate Hopf Bifurcation in Retarded Functional Differential Equations, and Applications to Endemic Bubbles

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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.

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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.

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Correspondence to Victor G. LeBlanc.

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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:

$$\begin{aligned}&K_1=\frac{1}{(1-\alpha ^*\tau )^2+\omega ^{*2}}\left[ 3(1-\alpha ^*\tau )f_{(3,0)} +\left( {\frac{3\,{\alpha ^*}^{2}\tau -{\alpha ^*}^{2}+{\beta ^*}^{2}-3\,\alpha ^*}{\beta ^*}}\right) f_{(2,1)}\right. \\&\quad -\,\left( {\frac{2\,{\alpha ^*}^{3}\tau +\alpha ^*\,{\beta ^*}^{2}\tau -2\,{\alpha ^*}^{3}+2 \,\alpha ^*\,{\beta ^*}^{2}-2\,{\alpha ^*}^{2}-{\beta ^*}^{2}}{{\beta ^*}^{2}}}\right) f_{(1,2)} +3\left( \,{\frac{{\alpha ^*}^{2}\tau -{\alpha ^*}^{2}+{\beta ^*}^{2}-\alpha ^*}{\beta ^*}}\right) f_{(0,3)}\\&\quad +\,2\left( \,{\frac{6\,{\alpha ^*}^{2}\tau -9\,\alpha ^*\,\beta ^*\,\tau -2\,{\alpha ^*}^{2}+ 2\,{\beta ^*}^{2}-6\,\alpha ^*+9\,\beta ^*}{ \left( \alpha ^*+\beta ^* \right) \left( 4\,\alpha ^*-5\,\beta ^* \right) }}\right) \times f_{(2,0)}^2\\&\quad -\,\left( {\frac{18\,{\alpha ^*}^{3}\tau -33\,{\alpha ^*}^{2}\beta ^*\,\tau +9\,\alpha ^*\,{ \beta ^*}^{2}\tau -10\,{\alpha ^*}^{3}+7\,{\alpha ^*}^{2}\beta ^*+10\,\alpha ^*\,{ \beta ^*}^{2}-7\,{\beta ^*}^{3}-18\,{\alpha ^*}^{2}+33\,\alpha ^*\,\beta ^*-9\,{\beta ^* }^{2}}{ \left( \alpha ^*+\beta ^* \right) \left( 4\,\alpha ^*-5\,\beta ^* \right) \beta ^*}}\right) \\&\quad \times \, f_{(2,0)}f_{(1,1)}-\,2\left( \,{\frac{ \left( \alpha ^*-\beta ^* \right) \left( 6\,{\alpha ^*}^{2}\tau -9 \,\alpha ^*\,\beta ^*\,\tau -6\,{\alpha ^*}^{2}+\alpha ^*\,\beta ^*+7\,{\beta ^*}^{2}-6\, \alpha ^*+9\,\beta ^* \right) }{ \left( \alpha ^*+\beta ^* \right) \left( 4\, \alpha ^*-5\,\beta ^* \right) \beta ^*}}\right) f_{(2,0)}f_{(0,2)}\\&\quad + \left( {\frac{ \left( \alpha ^*-\beta ^* \right) \left( 4\,{\alpha ^*}^{3}\tau -10\,{ \alpha ^*}^{2}\beta ^*\,\tau +\alpha ^*\,{\beta ^*}^{2}\tau -4\,{\alpha ^*}^{3}+2\,{ \alpha ^*}^{2}\beta ^*+3\,\alpha ^*\,{\beta ^*}^{2}-3\,{\beta ^*}^{3}-4\,{\alpha ^*}^{2} +10\,\alpha ^*\,\beta ^*-{\beta ^*}^{2} \right) }{{\beta ^*}^{2} \left( \alpha ^*+ \beta ^* \right) \left( 4\,\alpha ^*-5\,\beta ^* \right) }}\right) \\&\quad \times \, f_{(1,1)}^2+\,\left( {\frac{8\,\tau \,{\alpha ^*}^{5}+8\,{\alpha ^*}^{4}\beta ^*\,\tau -32\,{\alpha ^*}^ {3}{\beta ^*}^{2}\tau +19\,{\alpha ^*}^{2}{\beta ^*}^{3}\tau -9\,\alpha ^*\,{\beta ^*}^ {4}\tau -8\,{\alpha ^*}^{5}-8\,{\alpha ^*}^{4}\beta ^*+36\,{\alpha ^*}^{3}{\beta ^*}^{ 2}}{{\beta ^*}^{3} \left( \alpha ^*+\beta ^* \right) \left( 4\,\alpha ^*-5\,\beta ^* \right) }}\right. \\&\quad +\,\left. {\frac{{\alpha ^*}^{2}{\beta ^*}^{3}-28\,\alpha ^*\,{\beta ^*}^{4}+7\,{\beta ^*}^{5} -8\,{\alpha ^*}^{4}-8\,{\alpha ^*}^{3}\beta ^*+32\,{\alpha ^*}^{2}{\beta ^*}^{2}-19\, \alpha ^*\,{\beta ^*}^{3}+9\,{\beta ^*}^{4}}{{\beta ^*}^{3} \left( \alpha ^*+\beta ^* \right) \left( 4\,\alpha ^*-5\,\beta ^* \right) }}\right) f_{(1,1)}f_{(0,2)}\\&\quad -\,\, 2\left( \frac{4\alpha ^{*4}\tau +4\alpha ^{*3}\beta ^*\tau -13\alpha ^{*2}\beta ^{*2}\tau +2\alpha ^*\beta ^{*3}\tau -4\alpha ^{*4}-4\alpha ^{*3}\beta ^*+15\alpha ^{*2}\beta ^{*2}+4\alpha ^*\beta ^{*3}}{\beta ^{*2}(\alpha ^*+\beta ^*)(4\alpha ^*-5\beta ^*)}\right. \\&\quad +\, \left. \left. \frac{-11\beta ^{*4}-4\alpha ^{*3}-4\alpha ^{*2}\beta ^*+13\alpha ^*\beta ^{*2}-2\beta ^{*3}}{\beta ^{*2}(\alpha ^*+\beta ^*)(4\alpha ^*-5\beta ^*)}\right) \,f_{(0,2)}^2\right] . \end{aligned}$$

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.

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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

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