On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities

Llibre, Jaume; Ponce, Enrique; Torres, Francisco

doi:10.1088/0951-7715/21/9/013

Cited by 62 publications

(57 citation statements)

References 25 publications

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“…This situation was already considered in [15] but only when a R < 0 < a L and γ R γ L = 0. Here, we present new results including all the possible situations.…”

Section: Then the Change Of Variables (Different For Each Half-plane)mentioning

confidence: 95%

“…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%

“…For instance, in [9] the two linear systems share the linear part; in [10] both linear parts are simultaneously in real Jordan canonical form of focus type, which is a nongeneric situation; in [11] the two involved linear systems share the same equilibrium point; and in [15] only Liénard systems with neither real equilibria nor sliding set were considered.…”

Section: Introductionmentioning

confidence: 99%

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Canonical Discontinuous Planar Piecewise Linear Systems

Freire¹,

Ponce²,

Torres³

2012

SIAM J. Appl. Dyn. Syst.

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188

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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“…This situation was already considered in [15] but only when a R < 0 < a L and γ R γ L = 0. Here, we present new results including all the possible situations.…”

Section: Then the Change Of Variables (Different For Each Half-plane)mentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Canonical Discontinuous Planar Piecewise Linear Systems

Freire¹,

Ponce²,

Torres³

2012

SIAM J. Appl. Dyn. Syst.

Self Cite

188

142

View full text Add to dashboard Cite

show abstract

“…Due to their nonsmoothness, these systems can have richer dynamical phenomena than the smooth ones (see [3][4][5] and the references cited therein). For instance, nonsmooth system can have the sliding phenomena and in [6,7] Giannakopoulos and Pliete have studied the existence of sliding cycles and sliding homoclinic cycles for a planar relay control feedback systems.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the Number of Limit Cycles of a Piecewise Quadratic Near-Hamiltonian System

Tian

2014

Abstract and Applied Analysis

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“…This is, the creation or destruction of one crossing limit cycle occurs when a sliding segment changes its stability, this phenomenon is presented without demonstration in [18] and called pseudo-Hopf bifurcation. The appearance of a crossing limit cycle may occur in cases where there is not sliding segment, see [9,21,25].…”

Section: Introductionmentioning

confidence: 99%