The generic unfolding of a codimension-two connection to a two-fold singularity of planar Filippov systems

Abstract

Generic bifurcation theory was classically well developed for smooth differential systems, establishing results for k-parameter families of planar vector fields. In the present study we focus on a qualitative analysis of 2-parameter families, Z α,β , of planar Filippov systems assuming that Z 0,0 presents a codimension-two minimal set. Such object, named elementary simple two-fold cycle, is characterized by a regular trajectory connecting a visible two-fold singularity to itself, for which the second derivative of the first return map is nonvanishing. We analyzed the codimension-two scenario through the exhibition of its bifurcation diagram.2010 Mathematics Subject Classification. 34A36, 34C23, 37G15.

Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones

“…The present paper pays special attention to a class of critical crossing cycles, which are codimension-2 closed orbits composed by regular orbits and tangency points, in a two-parameter family of planar piecewise smooth system with two zones. Similar studies have been studied by several authors in the literatures [23,1,21,20,12,19] and the references cited therein. Precisely, [20] discussed the stability of a class of isolated homoclinic-like loops, which connecting a regular-fold point or a fold-fold point to itself, in planar piecewise smooth systems.…”

“…And then in the work of [21], authors provided some sufficient conditions to prove that there are at most one or two limit cycles can bifurcate from a generalized homoclinic loop. The complete bifurcation diagram of a closed trajectory with a fold-fold point is described in [23], and a two-parameter family of piecewise mechanical systems are provided to realized the results. The literature [1] developed a method to investigate crossing cycles bifurcating from a polycycle and applied it to describe the complete bifurcation diagram of vector fields around that polycycles.…”

Stability, cyclicity and bifurcation diagrams of polycycles in non-smooth planar vector fields

“…The next two theorems gives the bifurcation diagram of a polycycle composed by a hyperbolic saddle and a quadratic-regular tangential singularity. We observe that in the literature we already have the results of K. Andrade, O. Gomide and D. Novaes [1], D. Novaes, M. Teixeira and I. Zeli [15] and A. Mourtada [14], about the bifurcation diagrams of polycycles with two quadratic-regular tangential singularities, a quadratic-quadratic tangential singularities or two hyperbolic saddles, respectively. Theorem 4.…”

“…and thus from (15) we have the result. The case r n < 1 follows similarly from the fact that the inverse F −1 has order r −1 n in u. Corollary 2.…”

“…See Figures 3. For the bifurcation diagram of polycycles with two quadratic-tangential singularities or a quadratic-quadratic tangential singularity, see [1,15]. Now, for the bifurcation diagram of the polycycle given by two hyperbolic saddles, see [7,14].…”