Limit cycles bifurcating from planar polynomial quasi-homogeneous centers

Giné, Jaume; Grau, Maite; Llibre, Jaume

doi:10.1016/j.jde.2015.08.014

Cited by 13 publications

(14 citation statements)

References 9 publications

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“…Li et al [20] provided an upper bound for the number of limit cycles bifurcating from the period annulus of any homogeneous and quasi-homogeneous centers. Following this line Gavrilov et al [16] and Giné et al [17] gave some significant improvements of the result. In 1997, Cima et al [11] investigated the limit cycles for (1) in some special type.…”

Section: Introduction and Statements Of Main Resultsmentioning

confidence: 87%

A monodromy criterion for a type of degenerate system defined by the sum of two homogeneous vector fields

Huang¹,

Zhang²

2017

Rocky Mountain J. Math.

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In this paper we consider the limit cycles of the planar system d dt (x, y) = Xn + Xm,where Xn and Xm are quasi-homogeneous vector fields of degree n and m respectively. We prove that under a new hypothesis, the maximal number of limit cycles of the system is 1. We also show that our result can be applied to some systems when the previous results are invalid. The proof is based on the investigations for the Abel equation and the generalized-polar equation associated with the system, respectively. Usually these two kinds of equations need to be dealt with separately, and for both equations, an efficient approach to estimate the number of periodic solutions is constructing suitable auxiliary functions. In the present paper we develop a formula on the divergence, which allows us to construct an auxiliary function of one equation with the auxiliary function of the other equation, and vice versa.

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Section: Introduction and Statements Of Main Resultsmentioning

confidence: 87%

A monodromy criterion for a type of degenerate system defined by the sum of two homogeneous vector fields

Huang¹,

Zhang²

2017

Rocky Mountain J. Math.

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“…Li et al [21] provided an upper bound for the number of limit cycles bifurcating from the period annulus of any homogeneous and quasi-homogeneous centers. Following this line Gavrilov et al [16] and Giné et al [17] gave some significant improvements of the result. In 1997, Cima et al [11] investigated the limit cycles for (1) in some special type.…”

mentioning

confidence: 87%

“…As we know, system (1) in various types have been extensively studied and gained wide attention in decades [9,13,8,13,14,15,21,16,17,11,15,5,24,25,27,28,19]. One of the particularities of this system is that each limit cycle surrounding the origin can be expressed in generalized polar coordinates as r = r(θ), with r(θ) being a smooth periodic function, see for instance [8,13,14,15,19], etc.…”

mentioning

confidence: 99%

Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields

Huang¹,

Liang²

2021

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper we consider the limit cycles of the planar system d dt (x, y) = Xn + Xm,where Xn and Xm are quasi-homogeneous vector fields of degree n and m respectively. We prove that under a new hypothesis, the maximal number of limit cycles of the system is 1. We also show that our result can be applied to some systems when the previous results are invalid. The proof is based on the investigations for the Abel equation and the generalized-polar equation associated with the system, respectively. Usually these two kinds of equations need to be dealt with separately, and for both equations, an efficient approach to estimate the number of periodic solutions is constructing suitable auxiliary functions. In the present paper we introduce a formula on the divergence, which allows us to construct an auxiliary function of one equation with the auxiliary function of the other equation, and vice versa. Contents 1. Introduction and statements of main results 862 2. Preliminaries 864 2.1. A basic result for 1-dimensional non-autonomous differential equation 864 2.2. An estimate for the number of limit cycles of Abel equation 865 2.3. A formula on the divergence 867 2.4. The distribution of limit cycles of system (1) 869

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“…The centers of the quasi-homogeneous polynomial differential systems have been classified, and all are global centers see [15] and [19]. The normal forms of the quasi-homogeneous polynomial differential systems of degree n having a center at the origin were obtained in [34].…”

mentioning

confidence: 99%

“…Making a quasi-homogeneous blow-up x = r s1 cos θ, y = r s2 sin θ, which was also used in [15], we transform system (2) into the system…”

mentioning

confidence: 99%

Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems

Chen¹,

Llibre²,

Tang³

2019

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper we investigate the center problem for the discontinuous piecewise smooth quasi-homogeneous but non-homogeneous polynomial differential systems. First, we provide sufficient and necessary conditions for the existence of a center in the discontinuous piecewise smooth quasihomogeneous polynomial differential systems. Moreover, these centers are global, and the period function of their periodic orbits is monotonic. Second, we characterize the centers of the discontinuous piecewise smooth quasihomogeneous cubic and quartic polynomial differential systems.

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Limit cycles bifurcating from planar polynomial quasi-homogeneous centers

Abstract: Abstract. In this paper we find an upper bound for the maximum number of limit cycles bifurcating from the periodic orbits of any planar polynomial quasi-homogeneous center, which can be obtained using first order averaging method. This result improves the upper bounds given in [7].

Cited by 13 publications

References 9 publications

A monodromy criterion for a type of degenerate system defined by the sum of two homogeneous vector fields

A monodromy criterion for a type of degenerate system defined by the sum of two homogeneous vector fields

Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields

Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems

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