scholarly journals On the Closing Lemma for planar piecewise smooth vector fields

2016 ◽  
Vol 106 (6) ◽  
pp. 1174-1185 ◽  
Author(s):  
Tiago de Carvalho
Keyword(s):  
Vector Fields ◽  
Piecewise Smooth ◽  
Closing Lemma ◽  
Smooth Vector ◽  
Piecewise Smooth Vector Fields

Normal Forms for Codimension One Planar Piecewise Smooth Vector Fields

2014 ◽  
Vol 24 (07) ◽  
pp. 1450090 ◽  
Author(s):  
Tiago de Carvalho ◽  
Durval José Tonon
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Normal Forms ◽  
Piecewise Smooth ◽  
Topological Equivalence ◽  
Smooth Vector ◽  
Codimension One ◽  
Piecewise Smooth Vector Fields ◽  
Piecewise Smooth Vector Field

In this paper, we are dealing with piecewise smooth vector fields in a 2D-manifold. In such a scenario, the main goal of this paper is to exhibit the homeomorphism that gives the topological equivalence between a codimension one piecewise smooth vector field and the respective C0-normal form.

scholarly journals Birth of Isolated Nested Cylinders and Limit Cycles in 3D Piecewise Smooth Vector Fields with Symmetry

2020 ◽  
Vol 30 (07) ◽  
pp. 2050098
Author(s):  
Tiago Carvalho ◽  
Bruno Rodrigues de Freitas
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Vector Fields ◽  
Piecewise Smooth ◽  
Initial Model ◽  
Smooth Vector ◽  
Piecewise Smooth Vector Fields ◽  
Piecewise Smooth Vector Field

Our start point is a 3D piecewise smooth vector field defined in two zones and presenting a shared fold curve for the two smooth vector fields considered. Moreover, these smooth vector fields are symmetric relative to the fold curve, giving rise to a continuum of nested topological cylinders such that each orthogonal section of these cylinders is filled by centers. First, we prove that the normal form considered represents a whole class of piecewise smooth vector fields. After we perturb the initial model in order to obtain exactly [Formula: see text] invariant planes containing centers, a second perturbation of the initial model is also considered in order to obtain exactly [Formula: see text] isolated cylinders filled by periodic orbits. Finally, joining the two previous bifurcations we are able to exhibit a model, preserving the symmetry relative to the fold curve, and having exactly [Formula: see text] limit cycles.

scholarly journals ON THREE-PARAMETER FAMILIES OF FILIPPOV SYSTEMS — THE FOLD–SADDLE SINGULARITY

2012 ◽  
Vol 22 (12) ◽  
pp. 1250291 ◽  
Author(s):  
CLAUDIO AGUINALDO BUZZI ◽  
TIAGO DE CARVALHO ◽  
MARCO ANTONIO TEIXEIRA
Keyword(s):  
Vector Fields ◽  
Piecewise Smooth ◽  
Bifurcation Diagrams ◽  
Smooth Vector ◽  
Filippov Systems ◽  
Piecewise Smooth Vector Fields

This paper presents results concerning bifurcations of 2D piecewise-smooth vector fields. In particular, the generic unfoldings of codimension-three fold–saddle singularities of Filippov systems, where a boundary-saddle and a fold coincide, are considered and the bifurcation diagrams exhibited.

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

2014 ◽  
Vol 36 (2) ◽  
pp. 458-469 ◽  
Author(s):  
CLAUDIO A. BUZZI ◽  
TIAGO DE CARVALHO ◽  
RODRIGO D. EUZÉBIO
Keyword(s):  
Vector Fields ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Deterministic Chaos ◽  
Piecewise Smooth ◽  
Smooth Vector ◽  
Minimal Sets ◽  
Piecewise Smooth Vector Fields

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.

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