On a Żołądek theorem

Bondar, Yu. L.; Садовский, А. П.

doi:10.1134/s0012266108020158

Cited by 15 publications

(18 citation statements)

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“…In 1995In ,Żo ladek [1995 first used a rational Darboux integral and Melnikov functions up to second-order to claim the existence of 11 small limit cycles around a center. After more than ten years, another two cubic systems were constructed to show 11 limit cycles [Christopher, 2006;Bondar & Sadovskii, 2008]. The system considered in [Żo ladek, 1995] was reinvestigated by Yu and Han [2011] using the method of focus value computation, and only nine small limit cycles were obtained.…”

Section: Introductionmentioning

confidence: 99%

An Improvement on the Number of Limit Cycles Bifurcating from a Nondegenerate Center of Homogeneous Polynomial Systems

Han

2018

Int. J. Bifurcation Chaos

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In the two articles in Appl. Math. Comput., J. Giné [2012a, 2012b] studied the number of small limit cycles bifurcating from the origin of the system: [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are homogeneous polynomials of degree [Formula: see text]. It is shown that the maximal number of the small limit cycles, denoted by [Formula: see text], satisfies [Formula: see text] for [Formula: see text]; and [Formula: see text], [Formula: see text]. It seems that the correct answer for their case [Formula: see text] should be [Formula: see text]. In this paper, we apply Hopf bifurcation theory and normal form computation, and perturb the isolated, nondegenerate center (the origin) to prove that [Formula: see text] for [Formula: see text]; and [Formula: see text] for [Formula: see text], which improve Giné’s results with one more limit cycle for each case.

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Section: Introductionmentioning

confidence: 99%

An Improvement on the Number of Limit Cycles Bifurcating from a Nondegenerate Center of Homogeneous Polynomial Systems

Han

2018

Int. J. Bifurcation Chaos

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“…Christopher in [5] gave a simpler proof of Zoladek's result perturbing a Darboux cubic center. The same lower bound was also done with a different Darboux cubic center by Bondar and Sadovskii in [4]. By perturbations of a family of Darboux quartic (resp.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 68%

“…For general cubic system, this maximum number would be larger than or equal to 11, see [4,5,27,28]. For n = 4 and 5 the maximal order of a weak-focus is not less than 21 and 28 respectively, see [11,15].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Weak-Foci of High Order and Cyclicity

Liang

Torregrosa

2016

Qual. Theory Dyn. Syst.

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Abstract. A particular version of the 16th Hilbert's problem is to estimate the number, M (n), of limit cycles bifurcating from a singularity of center-focus type. This paper is devoted to finding lower bounds for M (n) for some concrete n by studying the cyclicity of different weak-foci. Since a weak-focus with high order is the most current way to produce high cyclicity, we search for systems with the highest possible weak-focus order. For even n, the studied polynomial system of degree n was the one obtained by [20] where the highest weak-focus order is n 2 + n − 2 for n = 4, 6, . . . , 18. Moreover, we provide a system which has a weak-focus with order (n − 1) 2 for n ≤ 12. We show that Christopher's approach [5], aiming to study the cyclicity of centers, can be applied also to the weak-focus case. We also show by concrete examples that, in some families, this approach is so powerful and the cyclicity can be obtained in a simple computational way. Finally, using this approach, we obtain that M (6) ≥ 39, M (7) ≥ 34 and M (8) ≥ 63.

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“…For systems with an elementary focus, the best result obtained so far is nine limit cycles [Yu & Corless, 2009;Chen et al, 2013;Lloyd & Pearson, 2012]. On the other hand, for systems with a center, there are also a few results obtained in the past two decades [Żoladek, 1995;Yu & Han, 2011;Tian & Yu, 2016;Bondar & Sadovskii, 2008]. Recently, the existence of 12 small-amplitude limit cycles around a single singular point was proved by Yu and Tian [2014].…”

Section: Introductionmentioning

confidence: 99%

Twelve Limit Cycles in 3D Quadratic Vector Fields with Z3 Symmetry

Guo

2018

Int. J. Bifurcation Chaos

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This paper is concerned with the number of limit cycles bifurcating in three-dimensional quadratic vector fields with [Formula: see text] symmetry. The system under consideration has three fine focus points which are symmetric about the [Formula: see text]-axis. Center manifold theory and normal form theory are applied to prove the existence of 12 limit cycles with [Formula: see text]–[Formula: see text]–[Formula: see text] distribution in the neighborhood of three singular points. This is a new lower bound on the number of limit cycles in three-dimensional quadratic systems.

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On a Żołądek theorem

Cited by 15 publications

References 1 publication

An Improvement on the Number of Limit Cycles Bifurcating from a Nondegenerate Center of Homogeneous Polynomial Systems

An Improvement on the Number of Limit Cycles Bifurcating from a Nondegenerate Center of Homogeneous Polynomial Systems

Weak-Foci of High Order and Cyclicity

Twelve Limit Cycles in 3D Quadratic Vector Fields with Z3 Symmetry

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