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 Local Bifurcations of Reversible Piecewise Smooth Planar Dynamical Systems

2020 ◽  
pp. 1-15
Author(s):  
V. Sh. Roitenberg
Keyword(s):  
Dynamical System ◽  
Vector Field ◽  
Dynamical Systems ◽  
Singular Point ◽  
Vector Fields ◽  
Singular Points ◽  
Piecewise Smooth ◽  
Smooth Vector ◽  
Local Bifurcations ◽  
Piecewise Smooth Vector Fields

There are quite a few works, which consider local bifurcations of piecewise-smooth vector fields on the plane. A number of papers also studied  the local bifurcations of smooth vector fields on the plane that are reversible with respect to involution. In the paper, we introduce reversible dynamical systems defined by piecewise-smooth vector fields on the coordinate plane (x, y) for which the discontinuity line y = 0 coincides with the set of fixed points of the system involution. We consider the generic one-parameter perturbations of such a vector field. The bifurcations of the singular point O lying on this line are described in two cases. In the first case, the point O is a rough saddle of the smooth vector fields that coincide with a piecewise smooth vector field in the half-planes y > 0 and y < 0. The parameter can be chosen so that for parameter values less than or equal to zero, the dynamical system has a unique singular point with four hyperbolic sectors in a vicinity of the point O. For positive values of the parameter in the vicinity of the point O, there are three singular points, a quasi-centre and two saddles, the separatrixes of which form a simple closed contour that bounds the cell from closed trajectories. In the second case, O is a rough node of the corresponding vector fields. The parameter can be chosen so that for values of the parameter less than or equal to zero, the dynamical system has a unique singular point in a vicinity of the point O, and all other trajectories are closed. For positive values of the parameter in the vicinity of the point O, there are three singular points, two nodes and a quasi-saddle, whose two separatrixes go to the nodes.

scholarly journals On Singular "Semifocus" Type Point Bifurcations of Piecewise Smooth Dynamical System

2018 ◽  
pp. 57-70
Author(s):  
V. Sh. Roitenberg
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Piecewise Smooth ◽  
Dimensional Manifold ◽  
Smooth Vector ◽  
Weak Focus ◽  
Smooth Vector Field ◽  
Small Parameters ◽  
Line Of Discontinuity ◽  
Piecewise Smooth Vector Field

For the processes described by dynamical systems, closed trajectories of dynamical systems are in line with periodic oscillations. Therefore, there is a considerable interest in describing the bifurcations of the generation of closed trajectories from equilibrium when the parameters change. In typical one-parameter and two-parameter families of smooth dynamical systems on a plane, closed trajectories can be generated only from equilibrium – weak focus. In mathematical modeling in the theory of automatic control, in mechanics and in other applications, piecewise smooth dynamical systems are often used. For them, there are other bifurcations of the generation of closed trajectories from equilibrium. The paper describes one of them, which is a typical family of dynamical systems specified by a piecewise smooth vector field on a two-dimensional manifold depending on two small parameters. It is assumed that for zero values of the parameters the vector field has a singular point O on the line of discontinuity of the field, and the point O is stable; in one half-neighborhood of the point O the field coincides with a smooth vector field for which the point O is a weak focus with positive (negative) first Lyapunov value, and in the other half-neighborhood it coincides with a smooth vector field directed at the points of the line of discontinuity inside the first of the semi-neighborhoods. The paper describes bifurcations in the neighborhood of the point O as the parameters change, in particular, indicating the regions of the parameters for which the vector field has a stable closed trajectory.

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.

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