Averaging theory for discontinuous piecewise differential systems

Llibre, Jaume; Mereu, Ana Cristina; Novaes, Douglas D.

doi:10.1016/j.jde.2015.01.022

Cited by 119 publications

(87 citation statements)

References 18 publications

(17 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In the past few decades, many authors have been devoted to study the number of limit cycles of piecewise smooth differential systems with two zones separated by a switching line, see [3,4,7,9,13,15,17,21,22,25,27] and the references quoted therein. The ways used in the aforementioned works are Melnikov function established in [8,17] and averaging method developed in [18,19]. Recently, Yang and Zhao [28] used the Picard-Fuchs equation to calculate the first order Melnikov function of a kind of piecewise smooth differential systems with a switching line, which can reduce a lot of calculation work.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Limit cycles appearing from the perturbation of differential systems with multiple switching curves

Yang

2020

Chaos, Solitons & Fractals

View full text Add to dashboard Cite

This paper deals with the problem of limit cycle bifurcations for a piecewise near-Hamilton system with four regions separated by algebraic curves y = ±x 2 . By analyzing the obtained first order Melnikov function, we give an upper bound of the number of limit cycles which bifurcate from the period annulus around the origin under n-th degree polynomial perturbations. In the case n = 1, we obtain that at least 4 (resp. 3) limit cycles can bifurcate from the period annulus if the switching curves are y = ±x 2 (resp. y = x 2 or y = −x 2 ). The results also show that the number of switching curves affects the number of limit cycles.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

Limit cycles appearing from the perturbation of differential systems with multiple switching curves

Yang

2020

Chaos, Solitons & Fractals

View full text Add to dashboard Cite

show abstract

“…The results stated in this subsection on the averaging theory are valid for discontinuous piecewise vector fields defined in R n and are proved in [21], but we shall state them for our discontinuous piecewise polynomial vector field X ε written in polar coordinates as the differential equation (1).…”

Section: Appendix: Averaging Theory For Discontinuous Piecewise Diffementioning

confidence: 99%

“…In the next result we present a version of the averaging theory for discontinuous piecewise vector fields, that is proved in [21], adapted to our differential equation (1). We note that in [21] the averaging theory uses that the Brouwer degree of a function f in a neighborhood of a zero ρ of the function f (ρ) is non-zero, while here we substitute this condition saying that the zero ρ is simple (i.e.…”

Section: Appendix: Averaging Theory For Discontinuous Piecewise Diffementioning

confidence: 99%

“…We note that in [21] the averaging theory uses that the Brouwer degree of a function f in a neighborhood of a zero ρ of the function f (ρ) is non-zero, while here we substitute this condition saying that the zero ρ is simple (i.e. df dρ (ρ) = 0), because this last condition implies that the mentioned Brouwer degree is non-zero.…”

Section: Appendix: Averaging Theory For Discontinuous Piecewise Diffementioning

confidence: 99%

See 1 more Smart Citation

Limit cycles of discontinuous piecewise polynomial vector fields

Carvalho

Llibre

Tonon

2017

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…For smooth systems this problem was extensively studied at any dimension (see, for instance, the book [9] and the references therein). For nonsmooth systems some techniques to deal with this kind of problem have been recently developed (see, for instance, [18,19]). However, due to the difficulty in applying these last ideas in higher dimensional systems, it has been considered, in general, for planar systems (see, for instance, [2,20,22,23]).…”

Section: Introductionmentioning

confidence: 99%

Piecewise smooth dynamical systems: Persistence of periodic solutions and normal forms

Gouveia

Llibre

Novaes

et al. 2016

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

Abstract. We consider a n-dimensional piecewise smooth vector fields with two zones separated by a hyperplane Σ which admits an invariant hyperplane Ω transversal to Σ containing a period-annulus A fulfilled by crossing periodic solutions. For small discontinuous perturbations of these systems we develop a Melnikov-like function to control the persistence of periodic solutions contained in A. When n = 3 we provide normal forms in the piecewise linear case. Finally we apply the Melnikov-like function to study discontinuous perturbations of the normal forms established.

show abstract

Averaging theory for discontinuous piecewise differential systems

Cited by 119 publications

References 18 publications

Limit cycles appearing from the perturbation of differential systems with multiple switching curves

Limit cycles appearing from the perturbation of differential systems with multiple switching curves

Limit cycles of discontinuous piecewise polynomial vector fields

Piecewise smooth dynamical systems: Persistence of periodic solutions and normal forms

Contact Info

Product

Resources

About