scholarly journals On the Limit Cycles of a Class of Generalized Kukles Polynomial Differential Systems via Averaging Theory

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Amar Makhlouf ◽  
Amor Menaceur
Keyword(s):  
Limit Cycles ◽  
Second Order ◽  
Averaging Theory ◽  
Differential Systems ◽  
Polynomial Differential Systems

We apply the averaging theory of first and second order to a class of generalized Kukles polynomial differential systems to study the maximum number of limit cycles of these systems.

scholarly journals On the number of limit cycles of a class of polynomial differential systems

2012 ◽  
Vol 468 (2144) ◽  
pp. 2347-2360 ◽  
Author(s):  
Jaume Llibre ◽  
Clàudia Valls
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Upper Bound ◽  
Second Order ◽  
Averaging Theory ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Generalized Polynomial

We study the number of limit cycles of polynomial differential systems of the form where g 1 , f 1 , g 2 and f 2 are polynomials of a given degree. Note that when g 1 ( x )= f 1 ( x )=0, we obtain the generalized polynomial Liénard differential systems. We provide an accurate upper bound of the maximum number of limit cycles that the above system can have bifurcating from the periodic orbits of the linear centre , using the averaging theory of first and second order.

scholarly journals Limit Cycles of a Class of Polynomial Differential Systems Bifurcating from the Periodic Orbits of a Linear Center

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1346 ◽  
Author(s):  
Amor Menaceur ◽  
Salah Boulaaras ◽  
Salem Alkhalaf ◽  
Shilpi Jain
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Second Order ◽  
Averaging Theory ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
New Class ◽  
Extended Work

In this paper, we study the number of limit cycles of a new class of polynomial differential systems, which is an extended work of two families of differential systems in systems considered earlier. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of a center using the averaging theory of first and second order.

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.

Limit Cycles of Perturbed Cubic Isochronous Center via the Second Order Averaging Method

2014 ◽  
Vol 24 (03) ◽  
pp. 1450035 ◽  
Author(s):  
Shimin Li ◽  
Yulin Zhao
Keyword(s):  
Limit Cycles ◽  
Averaging Method ◽  
Second Order ◽  
Isochronous Center ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Period Annulus ◽  
Isochronous System ◽  
Cubic Polynomial

In this paper, we bound the number of limit cycles for a class of cubic reversible isochronous system inside the class of all cubic polynomial differential systems. By applying the averaging method of second order to this system, it is proved that at most eight limit cycles can bifurcate from the period annulus. Moreover, this bound is sharp.

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

2020 ◽  
Vol 30 (04) ◽  
pp. 2050051
Author(s):  
Jaume Llibre ◽  
Arefeh Nabavi ◽  
Marzieh Mousavi
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Quadratic Polynomial ◽  
Averaging Theory ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Polynomial Differential Systems ◽  
Seventh Order ◽  
Whole Class

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).

scholarly journals Centers and Limit Cycles of Generalized Kukles Polynomial Differential Systems: Phase Portraits and Limit Cycles

2020 ◽  
pp. 378-398
Author(s):  
Ahlam Belfar ◽  
Rebiha Benterki
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Averaging Theory ◽  
Sixth Order ◽  
Phase Portraits ◽  
Differential Systems ◽  
Polynomial Differential Systems

In this work, we give the seven global phase portraits in the Poincar´e disc of the Kukles differential system given by x˙ = −y, y˙ = x + ax8 + bx4y4 + cy8, where x, y ∈ R and a, b, c ∈ R with a2 + b2 + c2 ̸= 0. Moreover, we perturb these system inside all classes of polynomials of eight degrees, then we use the averaging theory up sixth order to study the number of limit cycles which can bifurcate from the origin of coordinates of the Kukles differential system

Sign in / Sign up

Export Citation Format

Share Document