The number of limit cycles of a quintic polynomial system with center

Atabaigi, Ali; Nyamoradi, Nemat; Zangeneh, Hamid R. Z.

doi:10.1016/j.na.2009.01.213

Cited by 12 publications

(5 citation statements)

References 7 publications

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“…Other similar studies are seen in [Atabaigi et al, 2009;Yao & Han, 2012;Chang & Han, 2013;Liu & Han, 2013;Llibre & Zhang, 2001] and references therein. We may find that most papers considered the limit cycles bifurcating from the periodic orbits of the following systemẋ…”

Section: Introductionsupporting

confidence: 74%

Bifurcation of Limit Cycles from a Quasi-Homogeneous Degenerate Center

Geng

Lian

2015

Int. J. Bifurcation Chaos

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In this paper, we deal with the following differential systeṁwhere p, q are positive integers, and P (x, y), Q(x, y) are real polynomials of degree n, we obtain an upper bound for the maximum number of limit cycles bifurcating from the period annulus of a quasi-homogeneous center, that is (n − 1)p 1 + (t + 1)q − 1 + 2rp 1 q 1 (q + 3) + 2tqrp 1 q 1 , where t = [n/2q] + 2, (p, q) = r(p 1 , q 1 ), p 1 and q 1 are coprime.

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Section: Introductionsupporting

confidence: 74%

Bifurcation of Limit Cycles from a Quasi-Homogeneous Degenerate Center

Geng

Lian

2015

Int. J. Bifurcation Chaos

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“…Many authors these last years have studied this last problem restricted to the centers of the quadratic polynomial differential systems, see for instance the a Figure 1. The phase portrait in the Poincaré disc of system (1).…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…The tools for studying the limit cycles bifurcating from the periodic orbits surrounding a center are the Poincaré return map (see for instance [3,20]), the Poincaré-Melnikov integrals (see for example [13,14]), the Abelian integrals (see [1,6,28]), the averaging theory (see for instance [2,7]), and the inverse integrating factor (see [11]). While the first three methods only provide the number of bifurcated limit cycles, the averaging method and method which use the inverse integrating factor can also give the shape of bifurcated limit cycles.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Limit Cycles Bifurcating from a Family of Reversible Quadratic Centers via Averaging Theory

Llibre

Nabavi

Mousavi

2020

Int. J. Bifurcation Chaos

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Consider the class of reversible quadratic systems [Formula: see text] with [Formula: see text]. These quadratic polynomial differential systems have a center at the point [Formula: see text] and the circle [Formula: see text] is one of the periodic orbits surrounding this center. These systems can be written into the form [Formula: see text] with [Formula: see text]. For all [Formula: see text] we prove that the averaging theory up to seventh order applied to this last system perturbed inside the whole class of quadratic polynomial differential systems can produce at most two limit cycles bifurcating from the periodic orbits surrounding the center (0,0) of that system. Up to now this result was only known for [Formula: see text] (see Li, 2002; Liu, 2012).

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“…Basically, four methods have been used to perform such studies and they are based on: the Poincaré return map (see for instance [7,8,18]), the Poincaré-Pontrjagin-Melnikov integrals or Abelian integrals that are equivalent in the plane (see [2,11,3,4,6,10,25,26]), the inverse integrating factor (see [12,13,14,24]), and the averaging method which in the plane is also equivalent to the Abelian integrals (see for instance [5,17,19]). …”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Limit cycles bifurcating from a perturbed quartic center

Coll¹,

Llibre²,

Prohens³

2011

Chaos, Solitons & Fractals

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Abstract. We consider the quartic centerẋ = −yf (x, y),ẏ = xf (x, y), with f (x, y) = (x+a)(y+b)(x+c) and abc = 0. Here we study the maximum number σ of limit cycles which can bifurcate from the periodic orbits of this quartic center when we perturb it inside the class of polynomial vector fields of degree n, using the averaging theory of first order. We prove that 4[(n − 1)/2] + 4 ≤ σ ≤ 5[(n − 1)/2 + 14, where [η] denotes the integer part function of η.

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The number of limit cycles of a quintic polynomial system with center

Cited by 12 publications

References 7 publications

Bifurcation of Limit Cycles from a Quasi-Homogeneous Degenerate Center

Bifurcation of Limit Cycles from a Quasi-Homogeneous Degenerate Center

Limit Cycles Bifurcating from a Family of Reversible Quadratic Centers via Averaging Theory

Limit cycles bifurcating from a perturbed quartic center

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