Minimal sets and chaos in planar piecewise smooth vector fields

Some aspects concerning chaos and minimal sets in discontinuous dynamical systems are addressed. The orientability dependence of trajectories sliding trough some variety is exploited and new phenomena emerging from this situation are highlighted. In particular, although chaotic flows and nontrivial minimal sets are not allowed for smooth vector fields in the plane, the existence of such objects for some classes of vector fields is verified. A characterization of chaotic flows in terms of orientable minimal sets is also provided. The main feature of the dynamical systems under study is related to the non uniqueness of trajectories in some zero measure region as well as the orientation of orbits reaching such region

Discussion on the limit cycles of the Lev Ginzburg equation by M. Bellamy and R.E. Mickens

Agraïments: The first author thanks CNPq, CAPES and FAPESP for the financial support. The second author is supported by FAPESP, grant 2010/18015- 6. The fourth author is partially supported by CNPq grant 304926/2009-4 and by FAPEMIG grant PPM-00204-11. All the authors except the second one are also supported by the joint project CAPES-MECD grant PHB-2009-0025- PC

O método do Avering via teoria do grau de Brouwer e aplicações

Nosso objetivo neste trabalho é estudar o método do averaging através do grau topológico de Brouwer e utilizá-lo para investigar o número de ciclos limites que bifurcam de uma singularidade do tipo centro quando perturbamos um sistema de equações diferenciais através de um pequeno parâmetro ε. Começaremos apresentando o método do averaging que aaprece na literatura clássica e algumas aplicações deste. Depois faremos uma breve discussão sobre o grau topológico de Brouwer, seguido do teorema do averaging que faz menção a este conceito. Finalmente, exibiremos algumas aplicações inéditas do método.The aim of this is to study the averaging method using the Brouwer topological degree in order to investigative the number of limit cycles that can bifurcate from a center type singularity when a differential systemas is perturbed by a small parameter ε. To this respect, initially, we present classical averaging method and some of its applications. So we introduce the Brouwer topological degree, followed by the averaging theorem. Finally, we show some original applications of the averaging method.Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq

Discussion on the limit cycles of the Lev Ginzburg equation by M. Bellamy and R.E. Mickens

Agraïments: The first author thanks CNPq, CAPES and FAPESP for the financial support. The second author is supported by FAPESP, grant 2010/18015- 6. The fourth author is partially supported by CNPq grant 304926/2009-4 and by FAPEMIG grant PPM-00204-11. All the authors except the second one are also supported by the joint project CAPES-MECD grant PHB-2009-0025- PC