Generic bifurcations of low codimension of planar Filippov Systems

Abstract

In this article some qualitative and geometric aspects of non-smooth dynamical systems theory are discussed. The main aim of\ud this article is to develop a systematic method for studying local(and global) bifurcations in non-smooth dynamical systems. Our results deal with the classification and characterization of generic codimension-2 singularities of planar Filippov Systems as well as the presentation of the bifurcation diagrams and some dynamical consequencesPeer ReviewedPostprint (published version

The pseudo-Hopf bifurcation for planar discontinuous piecewise linear differential systems

“…Since the three kinds of regions in Σ are relatively open, their boundaries are the called tangency points: q ∈ Σ such that c T f − (q) = 0 or c T f + (q) = 0 (see [12,18]). That is, points where one of the two vector fields is tangent to Σ.…”

“…In smooth systems there is a well known mechanism to search for the occurrence of limit cycles, the Hopf bifurcation theorem, see [13,19]. There are analogous results for piecewise smooth systems, for the case of continuous systems see for example [6,7,26,27], and for the case of discontinuous systems see [1,8,11,12,14,18]. In the discontinuous ones we can have more than one limit cycle, either all crossing cycles or including one sliding cycle, and in fact, the determination of the number of limit cycle has been the subject of several recent papers, see [2,3,4,10,15,16,17,20,22,23,24].…”

Bifurcations of equilibrium sets in mechanical systems with dry friction

“…in [16][17][18][19][20][21][22][23][24][25]. Bifurcations of limit cycles of discontinuous systems are studied using a return map; see [16,17,22,24].…”

“…Here, the dynamics is understood by following the trajectories that become tangent to a discontinuity boundary. Guardia et al present in [25] a generic classification of bifurcations with codimension one and two in planar differential inclusions. However, the special structure of differential inclusions describing mechanical systems with dry friction, which we will analyse in the present paper, is considered to be non-generic by Guardia et al.…”

“…In the class of systems studied in the present paper, equilibrium sets occur generically, and persist when physically relevant perturbations are applied. Due to the difference in allowed perturbations, the scenarios observed in the present paper are not considered in [25]. In [17,21], the authors study bifurcations of equilibria in continuous systems, that are not differentiable.…”