Degenerate Hopf Bifurcations in Discontinuous Planar Systems

Coll, Bartomeu; Gasull, Armengol; Prohens, R.

doi:10.1006/jmaa.2000.7188

Cited by 136 publications

(95 citation statements)

References 9 publications

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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]. In fact, in [12] authors conjectured that the maximum number of limit cycles for the class of systems under study is exactly two.…”

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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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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“…Here, we present new results including all the possible situations. Clearly, the origin is then the single sliding point and is always a topological focus, whose Lyapunov constants could be obtained through the analysis done in [3]; here we follow a different approach to capture not only local but also global information. Proposition 4.2 (stability of the origin in systems without sliding set).…”

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

confidence: 99%

“…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.

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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“…As far as we know for discontinuous quadratic differential systems only the center problem and the Hopf bifurcation has been studied partially, see [8,9,10].…”

Section: Introductionmentioning

confidence: 99%

Limit cycles for discontinuous quadratic differential systems with two zones

Llibre

Mereu

2014

Journal of Mathematical Analysis and Applications

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Abstract. In this paper we study the maximum number of limit cycles given by the averaging theory of first order for discontinuous differential systems, which can bifurcate from the periodic orbits of the quadratic isochronous centersẋ = −y + x 2 ,ẏ = x + xy andẋ = −y + x 2 − y 2 , y = x + 2xy when they are perturbed inside the class of all discontinuous quadratic polynomial differential systems with the straight line of discontinuity y = 0.Comparing the obtained results for the discontinuous with the results for the continuous quadratic polynomial differential systems, this work shows that the discontinuous systems have at least 3 more limit cycles surrounding the origin than the continuous ones.

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Degenerate Hopf Bifurcations in Discontinuous Planar Systems

Cited by 136 publications

References 9 publications

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

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

Canonical Discontinuous Planar Piecewise Linear Systems

Limit cycles for discontinuous quadratic differential systems with two zones

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