The qualitative analysis of a class of planar Filippov systems

Shui, Shuliang; Zhang, Xuyang; Li, Jingjing

doi:10.1016/j.na.2010.04.053

Cited by 27 publications

(15 citation statements)

References 14 publications

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“…After the pioneering book of Filippov [5], in the context of planar Filippov systems, some interesting works can be found in Coll et al [2], Giannakopoulos and Pliete [8], Han and Zhang [10], Huan and Yang [11], Freire et al [7], Kuznetsov et al [13], Llibre et al [14], Shui et al [16], and especially in the recent thorough work by Guardia et al [4] in this journal.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the critical crossing cycle bifurcation in planar Filippov systems

Freire

Ponce

Torres

2015

Journal of Differential Equations

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We consider planar piecewise smooth differential systems with a discontinuity line, and characterize the critical crossing cycle bifurcation (CC bifurcation), also termed as homoclinic connection to a fold, see the celebrated papers by Kuznetzov et al. (2003) [13] and by Guardia et al. (2011) [4].The new characterization of the CC bifurcation achieved generically assures the persistence of one limit cycle, which changes from sliding to crossing, preserving its stability character. Therefore, we clarify some previous misconceptions: the case CC 2 described in Kuznetzov et al. [13] is not possible. Thus, Proposition 10.1 of Guardia et al. [4], predicting a CC 2 bifurcation in a piecewise linear normal form unfolding the codimension-2 focus-fold bifurcation, is not completely correct. The complete unfolding contains a region where two crossing limit cycles coexist, colliding to disappear at a new bifurcation curve.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On the critical crossing cycle bifurcation in planar Filippov systems

Freire

Ponce

Torres

2015

Journal of Differential Equations

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“…In [17] particular Filippov systems were both linear dynamics are of saddle type were studied. The Hopf bifurcation in non-smooth systems was studied in [12], [3] and [19].…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

The discontinuous matching of two planar linear foci can have three nested crossing limit cycles

Freire¹,

Ponce²,

Torres³

2014

Publicacions Matemàtiques

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The existence and stability of limit cycles in discontinuous piecewise linear systems obtained by the aggregation of two linear systems of focus type and having only one equilibrium point is considered. By using an adequate canonical form with five parameters, a thorough study of some Poincaré maps is performed. Different bifurcations which are responsible for the appearance of crossing limit cycles are detected and parameter regions with none, one, two and three crossing limit cycles are found.In particular, a first analytical proof of the existence, for certain differential systems in the considered family, of at least three homotopic crossing limit cycles surrounding the equilibrium point, is included. This fact has recently been numerically discovered in a particular example by S.-M. Huan and X.-S. Yang [13].

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“…After the pioneering work of Filippov [7], one must cite in this context the papers of Coll, Gasull, and Prohens [3], Giannakopoulos and Pliete [9], Huan and Yang [11], Kuznetsov, Rinaldi, and Gragnani [13], Llibre, Ponce, and Torres [15], Shui, Zhang, and Li [18], and the recent thorough work of Guardia, Seara, and Teixeira [6], among others.…”

Section: Introductionmentioning

confidence: 99%

Canonical Discontinuous Planar Piecewise Linear Systems

Freire¹,

Ponce²,

Torres³

2012

SIAM J. Appl. Dyn. Syst.

184

142

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Abstract. The family of Filippov systems constituted by planar discontinuous piecewise linear systems with two half-plane linearity zones is considered. Under generic conditions that amount to the boundedness of the sliding set, some changes of variables and parameters are used to obtain a Liénard-like canonical form with seven parameters. This canonical form is topologically equivalent to the original system if one restricts one's attention to orbits with no points in the sliding set. Under the assumption of focus-focus dynamics, a reduced canonical form with only five parameters is obtained. For the case without equilibria in both open half-planes we describe the qualitatively different phase portraits that can occur in the parameter space and the bifurcations connecting them. In particular, we show the possible existence of two limit cycles surrounding the sliding set. Such limit cycles bifurcate at certain parameter curves, organized around different codimension-two Hopf bifurcation points. The proposed canonical form will be a useful tool in the systematic study of planar discontinuous piecewise linear systems, in which this paper is a first step.Key words. Filippov systems, normal form, limit cycle, piecewise linear differential systems AMS subject classifications. 34C05, 34C07, 37G15DOI. 10.1137/11083928X1. Introduction. Piecewise linear (PWL) systems constitute a class of differential systems which are widely used to model many real processes and different devices; see, for instance, [1,5,17]. The case of continuous PWL systems with two linearity regions separated by a straight line is the simplest possible configuration in piecewise smooth systems. Such a family of systems was completely studied in a previous paper [8], where in particular the existence of at most one limit cycle was established.Enforced by modern nonlinear engineering problems and mathematical biology (see [5]), some recent works in the literature on planar PWL systems deal with vector fields where continuity at the common boundary is not assumed. After the pioneering work of Filippov [7], one must cite in this context the papers of Coll, Gasull, and Prohens [3], Giannakopoulos and Pliete [9], Huan and Yang [11], Kuznetsov, Rinaldi, and Gragnani [13], Llibre, Ponce, and Torres [15], Shui, Zhang, and Li [18], and the recent thorough work of Guardia, Seara, and Teixeira [6], among others.As mentioned in the recent work of Huan and Yang [11], the study of discontinuous PWL systems is a difficult task because of the lack of a canonical form that can cope with a sufficiently broad class of systems, in contrast to what can be done for the continuous case; see [2]. In fact, only very particular cases are thoroughly analyzed in the available literature.

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The qualitative analysis of a class of planar Filippov systems

Cited by 27 publications

References 14 publications

On the critical crossing cycle bifurcation in planar Filippov systems

On the critical crossing cycle bifurcation in planar Filippov systems

The discontinuous matching of two planar linear foci can have three nested crossing limit cycles

Canonical Discontinuous Planar Piecewise Linear Systems

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