Limit Cycles Bifurcating from the Period Annulus of Quasi-Homogeneous Centers

Li, Weigu; Llibre, Jaume; Yang, Jiazhong; Zhang, Zhifen

doi:10.1007/s10884-008-9126-1

Cited by 51 publications

(69 citation statements)

References 36 publications

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“…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]. …”

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confidence: 85%

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Limit cycles bifurcating from planar polynomial quasi-homogeneous centers

Giné

Grau

Llibre

2015

Journal of Differential Equations

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confidence: 85%

“…Lemma 4 statement (ii) (see below) are characterized all the centers of the quasi-homogeneous systems. This characterization is well-known see for instance [7]. Moreover in [1] the authors provide an algorithm for obtaining the quasi-homogeneous systems with a given degree which is a combinatorial problem.…”

Section: Introductionmentioning

confidence: 99%

Limit cycles bifurcating from planar polynomial quasi-homogeneous centers

Giné

Grau

Llibre

2015

Journal of Differential Equations

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“…They also show that these bounds are the best possible using the Abelian integral method of first order. In particular, applying statement (c) of Theorem A of [10] with p = q = 1 and n = 3 it follows that with the Abelian integral method of first order it can be obtained for systems (3) at most 1 limit cycle. For more details on the Abelian integral method see the second part of the book [4].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 96%

“…In [10] the authors provide upper bounds for the maximum number of limit cycles bifurcating from the periodic orbits of any homogeneous and quasi-homogeneous center, which can be obtained using the Abelian integral method of first order. They also show that these bounds are the best possible using the Abelian integral method of first order.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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Cubic homogeneous polynomial centers

Llibre

2014

Publicacions Matemàtiques

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Abstract. First, doing a combination of analytical and algebraic computations, we determine by first time an explicit normal form depending only on three parameters for all cubic homogeneous polynomial differential systems having a center.After using the averaging method of first order we show that we can obtain at most 1 limit cycle bifurcating from the periodic orbits of the mentioned centers when they are perturbed inside the class of all cubic polynomial differential systems. Moreover, there are examples with 1 limit cycles.

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Global centers of a family of cubic systems

Appis,

Llibre

2024

Aequat. Math.

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Consider the family of polynomial differential systems of degree 3, or simply cubic systems $$ x' = y, \quad y' = -x + a_1 x^2 + a_2 xy + a_3 y^2 + a_4 x^3 + a_5 x^2 y + a_6 xy^2 + a_7 y^3, $$ x ′ = y , y ′ = - x + a 1 x 2 + a 2 x y + a 3 y 2 + a 4 x 3 + a 5 x 2 y + a 6 x y 2 + a 7 y 3 , in the plane $$\mathbb {R}^2$$ R 2 . An equilibrium point $$(x_0,y_0)$$ ( x 0 , y 0 ) of a planar differential system is a center if there is a neighborhood $$\mathcal {U}$$ U of $$(x_0,y_0)$$ ( x 0 , y 0 ) such that $$\mathcal {U} \backslash \{(x_0,y_0)\}$$ U \ { ( x 0 , y 0 ) } is filled with periodic orbits. When $$\mathbb {R}^2\setminus \{(x_0,y_0)\}$$ R 2 \ { ( x 0 , y 0 ) } is filled with periodic orbits, then the center is a global center. For this family of cubic systems Lloyd and Pearson characterized in Lloyd and Pearson (Comput Math Appl 60:2797–2805, 2010) when the origin of coordinates is a center. We classify which of these centers are global centers.

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Limit Cycles Bifurcating from the Period Annulus of Quasi-Homogeneous Centers

Cited by 51 publications

References 36 publications

Limit cycles bifurcating from planar polynomial quasi-homogeneous centers

Limit cycles bifurcating from planar polynomial quasi-homogeneous centers

Cubic homogeneous polynomial centers

Global centers of a family of cubic systems

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