Formas normais de campos vetoriais reversiveis

Orientador: Marco Antonio TeixeiraTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação CientificaResumo: O conceito de reversibilidade para campos vetoriais está ligado a uma involução. Mais precisamente, dada uma involução de classe C8, f : IRn, 0 ? IRn, 0 (f2 = id), nós dizemos que um campo vetorial, (C8), X sobre IRn é rp-reversível do tipo (n, k) se f*X = -X o f e o conjunto S = Fix(f) é uma subvariedade k-dimensional de IRn. Todo ponto crítico de X em S é chamado uma singularidade simétrica de XAbstract: The concept of reversibility of a vector.field is linked with an involution. More precisely, given a smooth (C8) involution f : IRn, 0 ? IRn, 0 (f2 = id), we say that a smooth germ vector field X defined on IRn, 0 is f-reversible of type (n, k) if f*X = -X o f and the fixed point set of f, S = Fix(f), is a k-dimensional submanifold. Each singular point of X in S is called a symmetric singular point of XDoutoradoDoutor em Matemátic

Fold-Saddle Bifurcation in Non-Smooth Vector Fields on the Plane

This paper presents results concerning bifurcations of 2D piecewise-smooth dynamical systems governed by vector fields. Generic three parameter families of a class of Non-Smooth Vector Fields are studied and its bifurcation diagrams are exhibited. Our main result describes the unfolding of the so called Fold-Saddle singularity

Hopf bifurcation in the full repressilator equations

In this paper, we prove that the full repressilator equations in dimension six undergo a supercritical Hopf bifurcation

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

Periodic orbits for a class of reversible quadratic vector field on R-3

For a class of reversible quadratic vector fields on R-3 we study the periodic orbits that bifurcate from a heteroclinic loop having two singular points at infinity connected by an invariant straight line in the finite part and another straight line at infinity in the local chart U-2. More specifically, we prove that for all n is an element of N, there exists epsilon(n) > 0 such that the reversible quadratic polynomial differential systemx = a(0) + a(1y) + a(3y)(2) + a(4Y)(2) + epsilon(a(2x)(2) + a(3xz)),y = b(1z) + b(3yz) + epsilon b(2xy),z = c(1y) +c(4az)(2) + epsilon c(2xz)in R-3, with a(0) 0, c(2) < a(2) and b(3) is not an element of (c(4), 4c(4)), for epsilon is an element of (0, epsilon(n)) has at least n periodic orbits near the heteroclinic loop. (c) 2007 Elsevier B.V. All rights reserved