Limit cycles of cubic polynomial differential systems with rational first integrals of degree 2

Llibre, Jaume; Lopes, Bruno D.; Moraes, Jaime Rezende de

doi:10.1016/j.amc.2014.11.029

Cited by 6 publications

(6 citation statements)

References 16 publications

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“…inside the class of all cubic polynomial differential systems were studied inside the more general articles [5,10,11].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…Poincaré compactification. Let X ∈ P n (R 2 ) be any planar vector field of degree n. The Poincaré compactified vector field p(X ) corresponding to X is an analytic vector on S 2 defined as follows (see, for instance [11] or Chapter 5 of [6]). Let S 2 = {y = (y 1 , y 2 , y 3 ) ∈ R 3 : y 2 1 + y 2 2 + y 2 3 = 1} (the Poincaré sphere) and T y S 2 be the tangent space to S 2 at point y.…”

Section: Singular Pointsmentioning

confidence: 99%

“…11)ρ = ρ cos θ(b cos3 θ + (1 + ρ 2 ) sin 3 θ), θ = − sin θ(b cos 3 θ + sin 3 θ).The zeros on ρ = 0 of the differential system (11) are located at θ 1 = 0, θ 2 = π, θ 3 = − arctan b + π. The corresponding linear part for system(11) at (0, θ j ) for j = 1, 2 is…”

mentioning

confidence: 99%

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Centers and limit cycles of polynomial differential systems of degree 4 via averaging theory

Benterki

Llibre

2017

Journal of Computational and Applied Mathematics

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In this paper we classify the phase portraits in the Poincaré disc of the centers of the generalized class of Kukles systemṡ x = −y,ẏ = x + ax 3 y + bxy 3 , symmetric with respect to the y-axis, and we study, using the averaging theory up to sixth order, the limit cycles which bifurcate from the periodic solutions of these centers when we perturb them inside the class of all polynomial differential systems of degree 4.

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“…inside the class of all cubic polynomial differential systems were studied inside the more general articles [5,10,11].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Singular Pointsmentioning

confidence: 99%

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Centers and limit cycles of polynomial differential systems of degree 4 via averaging theory

Benterki

Llibre

2017

Journal of Computational and Applied Mathematics

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“…To the best of our knowledge, many authors have investigated the limit cycles for the quadratic Hamiltonian systems and non-Hamiltonian integrable systems under polynomial perturbations (e.g., [4,6,7,10,11,12,13,15,19] and references therein). However, the studies on cubic and higher degree systems are relatively few (e.g., [1,2,12,14,16,18,20]).…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…Recently, Llibre et al also study limit cycles of cubic polynomial differential systems with rational first integrals of degree 2 under polynomial perturbations of degree 3 in [16] using averaging method. They give six families of cubic polynomial differential systems, denote by P k for k = 1, 2, .…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Abelian integrals and limit cycles for a class of cubic polynomial vector fields of Lotka–Volterra type with a rational first integral of degree two

Cen

Zhao

Liang

2015

Journal of Mathematical Analysis and Applications

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In this paper, we study the number of limit cycles which bifurcate from the periodic orbits of cubic polynomial vector fields of Lotka-Volterra type having a rational first integral of degree 2, under polynomial perturbations of degree n. The analysis is carried out by estimating the number of zeros of the corresponding Abelian integrals. Moreover, using Chebyshev criterion, we show that the sharp upper bound for the number of zeros of the Abelian integrals defined on each period annulus is 3 for n = 3. The simultaneous bifurcation and distribution of limit cycles for the system with two period annuli under cubic polynomial perturbations are considered. All configurations (u, v) with 0 ≤ u, v ≤ 3, u + v ≤ 5 are realizable.Mathematics Subject Classification: Primary 34C05, 34A34, 37C07.

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Limit cycles bifurcating from the periodic annulus of the weight-homogeneous polynomial centers of weight-degree 2

Llibre

Lopes

Moraes

2016

Applied Mathematics and Computation

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Abstract. We obtain an explicit polynomial whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of a family of cubic polynomial differential centers when it is perturbed inside the class of all cubic polynomial differential systems. The family considered is the unique family of weight-homogeneous polynomial differential systems of weight-degree 2 with a center.

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Limit cycles of cubic polynomial differential systems with rational first integrals of degree 2

Cited by 6 publications

References 16 publications

Centers and limit cycles of polynomial differential systems of degree 4 via averaging theory

Centers and limit cycles of polynomial differential systems of degree 4 via averaging theory

Abelian integrals and limit cycles for a class of cubic polynomial vector fields of Lotka–Volterra type with a rational first integral of degree two

Limit cycles bifurcating from the periodic annulus of the weight-homogeneous polynomial centers of weight-degree 2

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