Limit cycles of discontinuous piecewise polynomial vector fields

Agraïments: The first author is partially supported by a FAPESP/Brazil grant number 2014/02134-7, a CNPq-Brazil grant number 443302/2014-6 and CAPES grants 88881.030454/2013-01 (from the program CSF-PVE) and 1576689 (from the program PNPD). The second author is partially supported by the CAPES grant 88881.030454/2013-01 from the program CSF-PVE. D.J. Tonon is supported by grant #2012/10 26 7000 803, Goiás Research Foundation (FAPEG), PROCAD/CAPES grant 88881.068462/2014-01 and by CNPq-Brazil.When the first average function is non-zero we provide an upper bound for the maximum number of limit cycles bifurcating from the periodic solutions of the center x= -y((x^2 y^2)/2)^m and y= x((x^2 y^2)/2)^m with m 1, when we perturb it inside a class of discontinuous piecewise polynomial vector fields of degree n with k pieces. The positive integers m, n and k are arbitrary. The main tool used for proving our results is the averaging theory for discontinuous piecewise vector fields

Crossing periodic orbits via first integrals

We characterize the families of periodic orbits of two discontinuous piecewise differential systems in R3 separated by a plane using their first integrals. One of these discontinuous piecewise differential systems is formed by linear differential systems, and the other by nonlinear differential systems

Limit cycles of piecewise smooth differential equations on two dimensional torus

In this paper we study the limit cycles of some classes of piecewise smooth vector fields defined in the two dimensional torus. The piecewise smooth vector fields that we consider are composed by linear, Ricatti with constant coefficients and perturbations of these one, which are given in (3). Considering these piecewise smooth vector fields we characterize the global dynamics, studying the upper bound of number of limit cycles, the existence of non-trivial recurrence and a continuum of periodic orbits. We also present a family of piecewise smooth vector fields that posses a finite number of fold points and, for this family we prove that for any 2k number of limit cycles there exists a piecewise smooth vector fields in this family that presents k number of limit cycles and prove that some classes of piecewise smooth vector fields presents a non-trivial recurrence or a continuum of periodic orbits

Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields

In the present study, we consider planar piecewise linear vector fields with two zones separated by the straight line x = 0. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has centre, a real one for y0, and such that the real centre is a global centre. Then, working with a first-order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second-order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one adittional crossing limit cycle can appear. Furthermore, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov method, the Extended Chebyshev systems with positive accuracy, and the Bendixson transformation

Phippov systems in tridimensional manifolds

Orientador: Marco Antonio TeixeiraTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação CientificaResumo: Neste trabalho sistemas dinâmicos descontínuos em variedades tridimensionais são estudados. Descrevemos uma classe de tais sistemas que são localmente estruturalmente estáveis em uma vizinhança de uma singularidade típica. Exibimos nessa etapa uma sub-família de campos do tipo dobra-dobra que é estruturalmente estável. Introduzimos os conceitos de A e L-estabilidade, que são pequenas generalizações dos conceitos clássicos de estabilidade assintótica e estabilidade no sentido de Lyapunov, respectivamente. Através de formas normais para as famílias de campos descontínuos de codimensão zero e um, exibimos os subconjuntos de sistemas descontínuos que são A e L-estáveis em uma vizinhança da origem. Destacamos um dos principais objetos de estudo desse trabalho: a singularidade dobra-dobra caso elíptico (T-singularidade). Discutimos algumas propriedades de sua dinâmica como a A-estabilidade para campos do tipo dobra-dobra de codimensão zero, um e dois. Investigamos também a presença de alguns invariantes topológicos, como separatrizes e famílias de órbitas periódicas. Finalmente, analisamos os chamados sistemas com relê. Em especial um sistema com dois relês acoplados é discutido.Abstract: In this work non-smooth dynamical systems in IR are considered. We describe a class of such systems that are locally structurally stable around a typical singularity. One of our contributions is to exhibit within these class of fold-fold systems a subclass which is structural stable. We also introduce the concept of A and L-stability which generalizes the classical concept of asymptotic and Lyapunov stability, respectively. Using normal forms for families of non smooth dynamical systems of codimension zero and one we exhibited subsets of non smooth dynamical systems which are A and L-stable in a neighborhood of the origin. We emphasize that the main object of study within this work is the fold-fold singularity in the elliptical case (T-singularity). We discuss some of its dynamical properties such as A-stability for codimension zero, one and two systems. We also investigate the presence of topological invariants such as séparatrices and families of periodic orbits. Finally we analyze two coupled relay systems.DoutoradoSistemas DinamicosDoutor em Matemátic

Um estudo global de campos de vetores planares

Neste trabalho estudamos os campos de vetores planares semi-homogênios quadráticos e também os campos de vetores planares com duas retas paralelas invariantes pelo fluxo. Para cada dessas classes, obtemos uma classificação dos retratos de fase global no disco de Poincaré e apresentamos as respectivas formas normais. Dentre as técnicas utilizadas no desenvolvimento do trabalho destacamos a Compactificação de Poincaré e o Método do Blow-up.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP

Symmetric periodic orbits for the collinear charged 3-body problem

Agraïments: The first author is partially supported by a CAPES grant number 88881.030454/2013-01 from the program CSF-PVE. The second author is supported by grant#2012/10 26 7000 803, Goiás Research Foundation (FAPEG), PRO-CAD/CAPES grant 88881.0 68462/2014-01 and by CNPq-Brazil.In this paper we study the existence of periodic symmetric orbits of the 3-body problem when each body possess mass and an electric charge. The main technique applied in this study is the continuation method of Poincar\'e

The symmetric periodic orbits for the two-electron atom

We analyse the existence of periodic symmetric orbits of the classical helium atom. The results obtained shows that there exists six families of periodic orbits that can be prolonged from a continuum of periodic symmetric orbits. The main technique applied in this study is the continuation method of Poincaré