Reversible equivariant Hopf bifurcation

In this paper we study codimension-one Hopf bifurcation from symmetric equilibrium points in reversible equivariant vector fields. Such bifurcations are characterized by a doubly degenerate pair of purely imaginary eigenvalues of the linearization of the vector field at the equilibrium point. The eigenvalue movements near such a degeneracy typically follow one of three scenarios: splitting (from two pairs of imaginary eigenvalues to a quadruplet on the complex plane), passing (on the imaginary axis), or crossing (a quadruplet crossing the imaginary axis). We give a complete description of the behaviour of reversible periodic orbits in the vicinity of such a bifurcation point. For non-reversible periodic solutions. in the case of Hopf bifurcation with crossing eigenvalues. we obtain a generalization of the equivariant Hopf Theorem.

Fold, transcritical and pitchfork singularities for time-reversible systems

This paper deals with a class of singularly perturbed reversible planar vector fields around the origin where the normal hyperbolicity assumption is not assumed. We exhibit conditions for the existence of infinitely many periodic orbits and hetero-clinic cycles converging to singular orbits with respect to the Hausdorf distance. In addition, generic normal forms of such singularities are presented.

Singular perturbation problems for time-reversible systems

In this paper singularly perturbed reversible vector fields defined in R-n without normal hyperbolicity conditions are discussed. The main results give conditions for the existence of infinitely many periodic orbits and heteroclinic cycles converging to singular orbits with respect to the Hausdorff distance.

On the existence of infinite heteroclinic cycles in polynomial systems and its dynamic consequences

In this work we consider the dynamic consequences of the existence of infinite heteroclinic cycle in planar polynomial vector fields, which is a trajectory connecting two saddle points at infinity. It is stated that, although the saddles which form the cycle belong to infinity, for certain types of nonautonomous perturbations the perturbed system may present a complex dynamic behavior of the solutions in a finite part of the phase plane, due to the existence of tangencies and transversal intersections of their stable and unstable manifolds. This phenomenon might be called the chaos arising from infinity. The global study at infinity is made via the Poincare Compactification and the argument used to prove the statement is the Birkhoff-Smale Theorem. (c) 2004 WILEY-NCH Verlag GmbH & Co. KGaA, Weinheim.

Bifurcation of limit cycles from a centre in R-4 in resonance 1:N

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

QCD phase transitions from relativistic hadron models

The models of translationally invariant infinite nuclear matter in the relativistic mean field models are very interesting and simple, since the nucleon can connect only to a constant vector and scalar meson field. Can one connect these to the complicated phase transitions of QCD? For an affirmative answer to this question, one must consider models where the coupling contstants to the scalar and vector fields depend on density in a nonlinear way, since as such the models are not explicitly chirally invariant. Once this is ensured, indeed one can derive a quark condensate indirectly from the energy density of nuclear matter which goes to zero at large density and temperature. The change to zero condensate indicates a smooth phase transition. © Springer-Verlag 1996.

Reversible Hamiltonian Liapunov center theorem

We study the existence of periodic solutions in the neighbourhood of symmetric (partially) elliptic equilibria in purely reversible Hamiltonian vector fields. These are Hamiltonian vector fields with an involutory reversing symmetry R. We contrast the cases where R acts symplectically and anti-symplectically. In case R acts anti-symplectically, generically purely imaginary eigenvalues are isolated, and the equilibrium is contained in a local two-dimensional invariant manifold containing symmetric periodic solutions encircling the equilibrium point. In case R acts symplectically, generically purely imaginary eigenvalues are doubly degenerate, and the equilibrium is contained in two two-dimensional invariant manifolds containing nonsymmetric periodic solutions encircling the equilibrium point. In addition, there exists a three-dimensional invariant surface containing a two-parameter family of symmetric periodic solutions.

Limit cycles for singular perturbation problems via inverse integrating factor

In this paper singularly perturbed vector fields Xε defined in ℝ2 are discussed. The main results use the solutions of the linear partial differential equation XεV = div(Xε)V to give conditions for the existence of limit cycles converging to a singular orbit with respect to the Hausdorff distance. © SPM.

Generic bifurcation of refracted systems

In this article we discuss some qualitative and geometric aspects of non-smooth dynamical systems theory. Our goal is to study the diagram bifurcation of typical singularities that occur generically in one parameter families of certain piecewise smooth vector fields named Refracted Systems. Such systems has a codimension-one submanifold as its discontinuity set. © 2012 Elsevier Ltd.

On singular foliations on the solid torus

We study smooth foliations on the solid torus S1×D2 having S1×{0} and S1×∂D2 as the only compact leaves and S1×{0} as singular set. We show that all other leaves can only be cylinders or planes, and give necessary conditions for the foliation to be a suspension of a diffeomorphism of the disc. © 2013 Elsevier B.V.

Estudo de ciclos limites em sistemas diferenciais lineares por partes

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Ciclos limites de campos de vetores polinomiais cúbicos e quadráticos

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Campos de vetores lineares reversíveis equivariantes

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Conjuntos limite e bifurfações de campos de vetores suaves por partes no plano

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Análise global de sistemas quadráticos e cúbicos com duas circunferências não-concêntricas invariantes

Pós-graduação em Matematica Aplicada e Computacional - FCT