# Shilnikov bifurcations in the Hopf-zero singularity

The so-called Hopf-zero singularity consists in a vector field in $R^3$ having the origin as a critical point, with a zero eigenvalue and a pair of conjugate purely imaginary eigenvalues.

The so-called Hopf-zero singularity consists in a vector field in $R^3$ having the origin as a critical point, with a zero eigenvalue and a pair of conjugate purely imaginary eigenvalues.

We study the dynamics of planar diffeomorphisms having a unique fixed point that is a hyperbolic local saddle. We obtain sufficient conditions under which the fixed point is a global saddle. We also address the special case of D2- symmetric maps, for which we obtain a similar result for C1 homeomorphisms. Some applications to differential equations are also given.

This is joint work with B. Alarcón (UFF — Brasil) and S.B.S.D. Castro (CMUP) and is part of a project of using symmetries to obtain global stability result

We study bifurcations in area-preserving maps with homoclinic tangencies. We consider $C^r$-smooth maps ($r\geq 3$) having a saddle fixed point whose stable and unstable invariant manifolds have a quadratic or cubic tangency at the points of some homoclinic orbit and study bifurcations of periodic orbits near the homoclinic tangencies in closed area-preserving maps. In the case of a quadratic homoclinic tangency we prove the existence of cascades of generic elliptic periodic points for one and two parameter unfoldings.

We study the complex dynamics in analytic area-preserving maps in a neighbourhood of a resonant elliptic fixed point. We assume that the resonance is weak, i.e., the linear part is a rotation of an angle 2\pi where n\geq5. Normal form theory suggests that there is a flower with n petals which consists of points bi-asymptotic to the fixed point.

We show that the flower splits: there are parabolic stable and unstable complex manifolds and they do not intersect. We measure the splitting of the manifolds and relate it to the Stokes phenomenon.

The notion of structural stability is one of the oldest concepts in dynamical systems. Structurally stable diffeomorphisms are conjugate to all nearby diffeomorphisms and are known to be strongly related to dynamics with persistence of the shadowing property.

Game theory studies the interactions among agents that try to maximize their payoffs by choosing from a set of actions. When this interaction occurs over time, the agents can change their choice of action depending on what they believe is the best action at any given time. This process constitutes a learning mechanism and can be described by various dynamical systems. Different dynamical systems arise as a result of different forms of belief update.

The human immunodeficiency virus (HIV) affects 34 to 46 million people worldwide. Of these, about 4 to 5 million people are coinfected with hepatitis C virus (HCV). Coinfection adds more severity to the two diseases. HIV accelerates the progression of HCV in dually infected patients. Having a count of CD4+ T cells below 200 cells/mm3 increases the risk of severe liver disease . Moreover, there is a higher risk of cirrhosis, end-stage liver disease, hepato-carcinoma, and hepatic-related death.

We investigate the dynamics near a heteroclinic cycle between a hyperbolic equilibrium E and a hyperbolic periodic solution P such that both connections are of codimension one. Such a cycle can be seen as the center of a two-parameter bifurcation scenario and, depending on properties of the transition maps, we find different types of (chaotic) shift dynamics near the cycle. Through our study we further explore the bifurcation diagrams previously outlined by others.

Takens' last problem. Whether are there persistent classes of smooth dynamical systems such that the set of initial states which give rise to orbits with

historic behavior has positive Lebesgue measure?

Colli-Vargas' conjecture. Every two-dimensional diffeomorphism with homoclinic tangency can be approximated in the Cr-topology by diffeomorphisms

having non-trivial wandering domains.