Limit Cycles for a Class of Continuous and Discontinuous Cubic Polynomial Differential Systems

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

doi:10.1007/s12346-014-0109-9

Cited by 17 publications

(10 citation statements)

References 20 publications

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“…Recently, by using Picard-Fuchs equations and Chebyshev criterion, J. Yang and L. Zhao [17] got the exactly 5 and 6 limit cycles for S 1 and S 2 , respectively. For more, one is recommended to see [4,[9][10][11][12].…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

On the number of limit cycles for generic Lotka–Volterra system and Bogdanov–Takens system under perturbations of piecewise smooth polynomials

Sui

Yang

Zhao

2019

Nonlinear Analysis: Real World Applications

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In this paper, we consider the bifurcation of limit cycles for generic L-V system (ẋ = y + x 2 − y 2 ± 4 √ 3 xy,ẏ = −x + 2xy) and B-T system (ẋ = y,ẏ = −x + x 2 ) under perturbations of piecewise smooth polynomials with degree n. Here the switching line is y = 0. By using Picard-Fuchs equations, we bound the number of zeros of first order Melnikov function which controls the number of limit cycles bifurcating from the center. It is proved that the upper bounds of the number of limit cycles for generic L-V system and B-T system are respectively 36n − 65 (n ≥ 4), 37, 57, 93 (n = 1, 2, 3) and 12n + 6.

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

On the number of limit cycles for generic Lotka–Volterra system and Bogdanov–Takens system under perturbations of piecewise smooth polynomials

Sui

Yang

Zhao

2019

Nonlinear Analysis: Real World Applications

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“…where F 1 , F 2 , R 1 , R 2 and h are continuous functions, locally Lipschitz in the variable x, T -periodic in the variable t, and h is a C 1 function having 0 as a regular value. The results stated in [19] have been extensively used, see for instance the works [16,17,26,21,22]. In this paper we focus on the development and improvement of the averaging theory for studying periodic solutions of a much bigger class of discontinuous differential systems than in (1).…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Periodic solutions of discontinuous second order differential systems

Llibre

Teixeira²

2014

jsing

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Motivated by problems coming from different areas of the applied science we study the periodic solutions of the following differential systemwhen F 0 , F 1 , and R are discontinuous piecewise functions, and ε is a small parameter. It is assumed that the manifold Z of all periodic solutions of the unperturbed system x ′ = F 0 (t, x) has dimension n or smaller then n. The averaging theory is one of the best tools to attack this problem. This theory is completely developed when F 0 , F 1 and R are continuous functions, and also when F 0 = 0 for a class of discontinuous differential systems. Nevertheless does not exist the averaging theory for studying the periodic solutions of discontinuous differential system when F 0 = 0. In this paper we develop this theory for a big class of discontinuous differential systems.

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“…, 0). But,f 1 is a polynomial on variables r and z of degree n − 1 and the functions f ℓ+1 given in (17) are polynomials on variables r and z of degree n for each ℓ = 1, 2, . .…”

Section: Proofs Of Corollaries 3 Andmentioning

confidence: 99%

Birth of limit cycles for a class of continuous and discontinuous differential systems in (d+ 2)–dimension

2015

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Abstract. The orbits of the reversible differential systemẋ = −y,ẏ = x,ż = 0, with x, y ∈ R and z ∈ R d , are periodic with the exception of the equilibrium points (0, 0, z). We compute the maximum number of limit cycles which bifurcate from the periodic orbits of the systemẋ = −y,ẏ = x,ż = 0, using the averaging theory of first order, when this system is perturbed, first inside the class of all polynomial differential systems of degree n, and second inside the class of all discontinuous piecewise polynomial differential systems of degree n with two pieces, one in y > 0 and the other in y < 0. In the first case this maximum number is n d (n − 1)/2, and in the second is n d+1 .

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Limit Cycles for a Class of Continuous and Discontinuous Cubic Polynomial Differential Systems

Cited by 17 publications

References 20 publications

On the number of limit cycles for generic Lotka–Volterra system and Bogdanov–Takens system under perturbations of piecewise smooth polynomials

On the number of limit cycles for generic Lotka–Volterra system and Bogdanov–Takens system under perturbations of piecewise smooth polynomials

Periodic solutions of discontinuous second order differential systems

Birth of limit cycles for a class of continuous and discontinuous differential systems in (d+ 2)–dimension

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