scholarly journals Upper bounds for the number of limit cycles of some planar polynomial differential systems

2010 ◽  
Vol 27 (1) ◽  
pp. 217-229 ◽  
Author(s):  
Armengol Gasull ◽  
◽  
Hector Giacomini ◽  
Keyword(s):  
Limit Cycles ◽  
Upper Bounds ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Planar Polynomial Differential Systems

On the Number of Limit Cycles of Discontinuous Liénard Polynomial Differential Systems

2018 ◽  
Vol 28 (14) ◽  
pp. 1850175
Author(s):  
Fangfang Jiang ◽  
Zhicheng Ji ◽  
Yan Wang
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Second Order ◽  
Discontinuous Differential Equations ◽  
Differential Systems ◽  
Lower Upper ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Averaging Theorem

In this paper, we investigate the number of limit cycles for two classes of discontinuous Liénard polynomial perturbed differential systems. By the second-order averaging theorem of discontinuous differential equations, we provide several criteria on the lower upper bounds for the maximum number of limit cycles. The results show that the second-order averaging theorem of discontinuous differential equations can predict more limit cycles than the first-order one.

scholarly journals Centers and limit cycles for a generalized kukles differential systems of degree 8

2021 ◽  
Author(s):  
Loubna Damene ◽  
Rebiha Benterki
Keyword(s):  
Limit Cycles ◽  
Main Tool ◽  
Upper Bounds ◽  
First Integrals ◽  
Averaging Theory ◽  
Phase Portraits ◽  
Differential Systems ◽  
Polynomial Differential Systems

Abstract In this paper we provide all the global phase portraits of the generalized kukles differential systems x= y; y = x + ax8 + bx6y2 + cx4y4 + dx2y6 + ey8; symmetric with respect to the x{axis, with a2 + b2 + c2 + d2 + e2 6= 0, and by using the averaging theory up to seven order, we give the upper bounds of limit cycles which can bifurcate from its center when we perturb it inside the class of all polynomial differential systems of degree 8. The main tool used for proving these results is based in the first integrals of the systems which form the discontinuous piecewise differential systems.

scholarly journals Limit cycles of piecewise polynomial differential systems with the discontinuity line xy = 0

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tao Li ◽  
Jaume Llibre
Keyword(s):  
Periodic Orbits ◽  
Lower Bounds ◽  
Limit Cycles ◽  
Averaging Method ◽  
Polynomial Systems ◽  
Upper Bounds ◽  
Piecewise Polynomial ◽  
Differential Systems ◽  
Discontinuity Line ◽  
Polynomial Differential Systems

<p style='text-indent:20px;'>In this paper we study the maximum number of limit cycles bifurcating from the periodic orbits of the center <inline-formula><tex-math id="M1">\begin{document}$ \dot x = -y((x^2+y^2)/2)^m, \dot y = x((x^2+y^2)/2)^m $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ m\ge0 $\end{document}</tex-math></inline-formula> under discontinuous piecewise polynomial (resp. polynomial Hamiltonian) perturbations of degree <inline-formula><tex-math id="M3">\begin{document}$ n $\end{document}</tex-math></inline-formula> with the discontinuity set <inline-formula><tex-math id="M4">\begin{document}$ \{(x, y)\in\mathbb{R}^2: xy = 0\} $\end{document}</tex-math></inline-formula>. Using the averaging theory up to any order <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula>, we give upper bounds for the maximum number of limit cycles in the function of <inline-formula><tex-math id="M6">\begin{document}$ m, n, N $\end{document}</tex-math></inline-formula>. More importantly, employing the higher order averaging method we provide new lower bounds of the maximum number of limit cycles for several types of piecewise polynomial systems, which improve the results of the previous works. Besides, we explore the effect of 4-star-symmetry on the maximum number of limit cycles bifurcating from the unperturbed periodic orbits. Our result implies that 4-star-symmetry almost halves the maximum number.</p>

Picard–Fuchs Equation Applied to Quadratic Isochronous Systems with Two Switching Lines

2020 ◽  
Vol 30 (03) ◽  
pp. 2050042
Author(s):  
Jihua Yang
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Isochronous Systems

The present paper is devoted to study the problem of limit cycle bifurcations for nonsmooth integrable differential systems with two perpendicular switching lines. By using the Picard–Fuchs equation, we obtain the upper bounds of the number of limit cycles bifurcating from the period annuli of the quadratic isochronous systems [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text], when they are perturbed inside a class of all discontinuous polynomial differential systems of degree [Formula: see text]. This method can be applied to study the limit cycle bifurcations of other integrable differential systems.

Limit Cycles for a Class of Zp-Equivariant Differential Systems

2020 ◽  
Vol 30 (08) ◽  
pp. 2050115
Author(s):  
Jing Gao ◽  
Yulin Zhao
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Melnikov Function ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Number Of Zeros ◽  
Planar Polynomial Differential Systems

In this paper, we study a class of [Formula: see text]-equivariant planar polynomial differential systems [Formula: see text]. It is shown that for any [Formula: see text] there is a differential system of the above type having at least [Formula: see text] limit cycles. This is proved by estimating the number of zeros of the first-order Melnikov function.

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