A general mechanism to generate three limit cycles in planar Filippov systems with two zones

Freire, Emilio; Ponce, Enrique; Torres, Francisco

doi:10.1007/s11071-014-1437-7

Cited by 125 publications

(83 citation statements)

References 13 publications

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“…In general these differential systems are linear, see for instance [3,6,9,11,12,13,14,15,16,17,18,22,23,24,30]. But there are very few works of continuous and discontinuous piecewise differential systems with an arbitrary number k of pieces.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Limit cycles of discontinuous piecewise polynomial vector fields

Carvalho

Llibre

Tonon

2017

Journal of Mathematical Analysis and Applications

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Abstract. When the first average function is non-zero we provide an upper bound for the maximum number of limit cycles bifurcating from the periodic solutions of the centerẋ = −y((x 2 + y 2 )/2) m andẏ = x((x 2 + y 2 )/2) m with m ≥ 1, when we perturb it inside a class of discontinuous piecewise polynomial vector fields of degree n with k pieces. The positive integers m, n and k are arbitrary. The main tool used for proving our results is the averaging theory for discontinuous piecewise vector fields.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Limit cycles of discontinuous piecewise polynomial vector fields

Carvalho

Llibre

Tonon

2017

Journal of Mathematical Analysis and Applications

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“…Up to now we know that there are discontinuous systems with at least three limit cycles, see for instance [2,4,3,6,8,9,10,11,12,13,14,15,22,17,18,19,20,22,24].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…We note that the denominator of t + cannot be zero, otherwise t + will be infinity, and we cannot have a periodic solution solution satisfying (10). Also the denominator of y 0 cannot be zero, otherwise b = 0 and therefore x − (t) = ct.…”

Section: Discontinuous Piecewise Differential System: Proof Of Theoremmentioning

confidence: 94%

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Piecewise linear differential systems without equilibria produce limit cycles?

Llibre

Teixeira

2016

Nonlinear Dyn

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Abstract. In this article we study the planar piecewise differential systems formed by two linear differential systems separated by a straight line, such that both linear differential have no equilibria, neither real nor virtual.When the piecewise differential system is continuous, we show that the system has no limit cycles. But when the piecewise differential system is discontinuous, we show that it can have at most one limit cycle.

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“…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].…”

Section: Introductionmentioning

confidence: 96%