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

Abstract

Abstract. This paper is concerned with the local bifurcation analysis around typical singularities of piecewise smooth planar dynamical systems. Three−parameter families of a class of non−smooth vector fields are studied and the tridimensional bifurcation diagrams are exhibited. Our main results describe the unfolding of the so called f old − cusp singularity by means of the variation of 3 parameters.

On Poincaré-Bendixson Theorem and non-trivial minimal sets in planar nonsmooth vector fields

“…Finding limit sets of trajectories of vector fields is one of the most important tasks of the qualitative theory of dynamical systems. In the literature there are several recent papers (see for instance [3,4,6,7]) where the authors explicitly exhibit the phase portraits of some NSVFs with their unfoldings. However, all the limit sets exhibited have trivial minimal sets (i.e., the minimal sets are equilibria, pseudo equilibria, cycles or pseudo-cycles).…”

Asymptotic stability and bifurcations of 3D piecewise smooth vector fields

“…The main tool used in this paper is the theory of the contact between a vector field and the boundary of a manifold, since the traditional methods involving Lyapunov functions do not apply here, see [6][7][8]15,19,24].…”

Smoothing of homoclinic-like connections to regular tangential singularities in Filippov systems

“…In this work, we shall consider planar piecewise smooth vector fields. More precisely, consider M an open subset of R 2 and N Ă M a codimension 1 submanifold of M. For each i = 1, 2, . .…”

“…This method was applied to describe bifurcation diagrams of Filippov systems around several Σ-polycycles. The readers are referred to [1,2,7,13,15] for more studies on Σ´polycycles. The interest in studying polycycles is due to the fact that they are non-local invariant sets that provide information on the dynamics of the system.…”