In this paper we completely study bifurcations of an epidemic model with five parameters introduced by Hilker et al. (Am Nat 173:72-88, 2009), which describes the joint interplay of a strong Allee effect and infectious diseases in a single population. Existence of multiple positive equilibria and all kinds of bifurcation are examined as well as related dynamical behavior. It is shown that the model undergoes a series of bifurcations such as saddle-node bifurcation, pitchfork bifurcation, Bogdanov-Takens bifurcation, degenerate Hopf bifurcation of codimension two and degenerate elliptic type Bogdanov-Takens bifurcation of codimension three. Respective bifurcation surfaces in five-dimensional parameter spaces and related dynamical behavior are obtained. These theoretical conclusions confirm their numerical simulations and conjectures by Hilker et al., and reveal some new bifurcation phenomena which are not observed in Hilker et al. (Am Nat 173:72-88, 2009). The rich and complicated dynamics exhibit that the model is very sensitive to parameter perturbations, which has important implications for disease control of endangered species.

We propose in this paper a method for obtaining a significant refinement of normal forms for dynamical systems or vector fields, with concrete and interesting applications. We use lower order nonlinear terms in the normal form for the simplifications of higher order terms. Our approach is applicable for both the non nilpotent and the nilpotent cases. For dynamical systems of dimensions 2 and 3 we give an algorithm that leads to interesting finite order normal forms which are optimal (or unique) with respect to equivalence by formal near identity transformations. We can compute at the same time a formal diffeormorphism that realizes the normalization. Comparisons with other methods are given for several examples. Academic Press

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We study the problem of existence of homoclinic and heteroclinic orbits of Chen's system. For the case of 2c > a > c > 0 and b ≥ 2a, we prove that the system has no homoclinic orbit but has two and only two heteroclinic orbits.