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A degenerate Hopf bifurcation in retarded functional differential equations, and applications to endemic bubbles. (English) Zbl 1346.34049

The paper is concerned with the study of a degenerate Hopf bifurcation for a smoothly parametrized family \[ \dot{z}(t)=L(\lambda,\mu)z_t+F(z_t,\lambda,\mu)\tag{1} \] of scalar retarded functional differential equations. Here, \(\lambda\) and \(\mu\) are real-valued parameters, \(z_t\in C=C([-\tau,0],\mathbb{R})\), \(\tau>0\), is defined by \(z_t(\theta)=z(t+\theta)\) for all \(t\in[-\tau,0]\), \(L(\lambda,\mu)\) stands for a smoothly parametrized family of bounded linear operators from \(C\) into \(\mathbb{R}\), and \(F\) denotes a smooth (nonlinear) function from \(C\times\mathbb{R}\times\mathbb{R}\) into \(\mathbb{R}\) with \(F(0,\lambda,\mu)=0\) for all \(\lambda,\mu\) with \(|\lambda|,|\mu|\) sufficiently small. The author establishes sufficient conditions such that the zero solution of Eq. (1) undergoes a degenerate Hopf bifurcation at \(\lambda=\mu=0\) where the degeneracy arises from a violation of the eigenvalue crossing condition in the classical Hopf bifurcation theorem. This is done by using a center manifold reduction along with Poincaré-Birkhoff normal forms and a classification of degenerate Hopf bifurcations given by M. Golubitsky and W. F. Langford [J. Differ. Equations 41, 375–415 (1981; Zbl 0442.58020)].
As a special case of Eq. (1), a family of scalar differential equations \[ \dot{x}(t)=\alpha(\lambda,\mu)x(t)+\beta(\lambda,\mu)x(t-\tau)+f(x(t),x(t-\tau),\lambda,\mu) \] with constant delay \(\tau>0\) and real parameter-dependent coefficients \(\alpha(\lambda,\mu),\beta(\lambda,\mu)\) is considered, and the obtained theoretical results are applied to a SIS model developed and analyzed in [M. Liu et al., SIAM J. Appl. Math. 75, No. 1, 75–91 (2015; Zbl 1328.34085)]. In this way, the author particularly shows that the phenomenon of an endemic bubble (i.e., a branch of periodic solutions bifurcates from the endemic equilibrium at some value of the basic reproduction number \(R_0\) and then reconnects to the endemic equilibrium at some larger value of \(R_0\)), which was reported in [Liu, loc. cit.], is a consequence of a degenerate Hopf bifurcation.

MSC:

34K18 Bifurcation theory of functional-differential equations
92D30 Epidemiology
34K17 Transformation and reduction of functional-differential equations and systems, normal forms
34K19 Invariant manifolds of functional-differential equations
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