Normal Forms for Codimension One Planar Piecewise Smooth Vector Fields

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

Chaotic planar piecewise smooth vector fields with non-trivial minimal sets

In this paper some aspects on chaotic behavior and minimality in planar piecewise smooth vector fields theory are treated. The occurrence of non-deterministic chaos is observed and the concept of orientable minimality is introduced. Some relations between minimality and orientable minimality are also investigated and the existence of new kinds of non-trivial minimal sets in chaotic systems is observed. The approach is geometrical and involves the ordinary techniques of non-smooth systems.

On 3-Parameter Families of Piecewise Smooth Vector Fields in the Plane

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

Basin of attraction of a cusp-fold singularity in 3D piecewise smooth vector fields

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

STABLE PIECEWISE POLYNOMIAL VECTOR FIELDS

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

Closed poly-trajectories and Poincaré index of non-smooth vector fields on the plane

This paper is concerned with closed orbits of non-smooth vector fields on the plane. For a class of non-smooth vector fields we provide necessary and sufficient conditions for the existence of closed poly-trajectorie. By means of a regularization process we prove that hyperbolic closed poly-trajectories are limit sets of a sequence of limit cycles of smooth vector fields. In our approach the Poincaré Index for non-smooth vector fields is introduced. © 2013 Springer Science+Business Media New York.

Conjuntos minimais e caóticos em campos de vetores planares suaves por partes

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Slow-fast systems on algebraic varieties bordering piecewise-smooth dynamical systems

This article extends results contained in Buzzi et al. (2006) [4], Llibre et al. (2007, 2008) [12,13] concerning the dynamics of non-smooth systems. In those papers a piecewise C-k discontinuous vector field Z on R-n is considered when the discontinuities are concentrated on a codimension one submanifold. In this paper our aim is to study the dynamics of a discontinuous system when its discontinuity set belongs to a general class of algebraic sets. In order to do this we first consider F :U -> R a polynomial function defined on the open subset U subset of R-n. The set F-1 (0) divides U into subdomains U-1, U-2,...,U-k, with border F-1(0). These subdomains provide a Whitney stratification on U. We consider Z(i) :U-i -> R-n smooth vector fields and we get Z = (Z(1),...., Z(k)) a discontinuous vector field with discontinuities in F-1(0). Our approach combines several techniques such as epsilon-regularization process, blowing-up method and singular perturbation theory. Recall that an approximation of a discontinuous vector field Z by a one parameter family of continuous vector fields is called an epsilon-regularization of Z (see Sotomayor and Teixeira, 1996 [18]; Llibre and Teixeira, 1997 [15]). Systems as discussed in this paper turn out to be relevant for problems in control theory (Minorsky, 1969 [16]), in systems with hysteresis (Seidman, 2006 [17]) and in mechanical systems with impacts (di Bernardo et al., 2008 [5]). (C) 2011 Elsevier Masson SAS. All rights reserved.

Sliding Vector Fields via Slow-Fast Systems

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Birth of limit cycles bifurcating from a nonsmooth center

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

ON THREE-PARAMETER FAMILIES of FILIPPOV SYSTEMS - THE FOLD-SADDLE SINGULARITY

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

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.

Resonances and subharmonic bifurcations of large amplitude periodic orbits of planar polynomial vector fields

In this work are studied periodic perturbations, depending on two parameters, of planar polynomial vector fields having an annulus of large amplitude periodic orbits, which accumulate on a symmetric infinite heteroclinic cycle. Such periodic orbits and the heteroclinic trajectory can be seen only by the global consideration of the polynomial vector fields on the whole plane, and not by their restriction to any compact set. The global study involving infinity is performed via the Poincare Compactification. It is shown that, for certain types of periodic perturbations, one can seek, in a neighborhood of the origin in the parameter plane, curves C-(m) of subharmonic bifurcations, for which the periodically perturbed system has subharmonics of order m, for any integer m.

Periodic perturbations of quadratic planar polynomial vector fields

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Sliding vector fields for non-smooth dynamical systems having intersecting switching manifolds

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)