On the number of limit cycles of the equation \frac{𝑑𝑦}𝑑π‘₯=\frac{𝑐π‘₯+𝑑𝑦+𝑃(π‘₯,𝑦)}π‘Žπ‘₯+𝑏𝑦+𝑄(π‘₯,𝑦), where 𝑃(π‘₯,𝑦) and 𝑄(π‘₯,𝑦) are homogeneous polynomials of degree 𝑛

10.1090/trans2/029/11 β—½ Β 
1963 β—½ Β 
pp. 289-293 β—½ Β 
Author(s): Β 
B. M. Peretjagin
Keyword(s): Β 
Limit Cycles β—½ Β 
Homogeneous Polynomials

scholarly journals EXPLICIT FORM FOR THE FIRST INTEGRAL AND LIMIT CYCLES OF A CLASS OF PLANAR KOLMOGOROV SYSTEMS

2021 β—½ Β 
Vol 55 (1 (254)) β—½ Β 
pp. 1-11
Author(s): Β 
Rachid Boukoucha
Keyword(s): Β 
Explicit Form β—½ Β 
Limit Cycles β—½ Β 
First Integral β—½ Β 
Homogeneous Polynomials

In this paper we characterize the integrability and the non-existence of limit cycles of Kolmogorov systems of the form \begin{equation*} \left\{ \begin{array}{l} x^{\prime }=x\left( R\left( x,y\right) \exp \left( \dfrac{A\left( x,y\right) }{B\left( x,y\right) }\right) +P\left( x,y\right) \exp \left( \dfrac{C\left( x,y\right) }{D\left( x,y\right) }\right) \right) , \\ \\ y^{\prime }=y\left( R\left( x,y\right) \exp \left( \dfrac{A\left( x,y\right) }{B\left( x,y\right) }\right) +Q\left( x,y\right) \exp \left( \dfrac{V\left( x,y\right) }{W\left( x,y\right) }\right) \right) , \end{array} \right. \end{equation*} where $A\left( x,y\right)$, $B\left( x,y\right)$, $C\left( x,y\right)$, $D\left( x,y\right)$, $P\left( x,y\right)$, $Q\left( x,y\right)$, $R\left(x,y\right)$, $V\left( x,y\right)$, $W\left( x,y\right)$ are homogeneous polynomials of degree $a$, $a$, $b$, $b$, $n$, $n$, $m$, $c$, $c$, respectively. Concrete example exhibiting the applicability of our result is introduced.

An Improvement on the Number of Limit Cycles Bifurcating from a Nondegenerate Center of Homogeneous Polynomial Systems

2018 β—½ Β 
Vol 28 (06) β—½ Β 
pp. 1850078 β—½ Β 
Author(s): Β 
Pei Yu β—½ Β 
Maoan Han β—½ Β 
Jibin Li
Keyword(s): Β 
Hopf Bifurcation β—½ Β 
Normal Form β—½ Β 
Limit Cycle β—½ Β 
Limit Cycles β—½ Β 
Correct Answer β—½ Β 
Homogeneous Polynomial β—½ Β 
Bifurcation Theory β—½ Β 
Polynomial Systems β—½ Β 
Homogeneous Polynomials β—½ Β 
Appl Math

In the two articles in Appl. Math. Comput., J. GinΓ© [2012a, 2012b] studied the number of small limit cycles bifurcating from the origin of the system: [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are homogeneous polynomials of degree [Formula: see text]. It is shown that the maximal number of the small limit cycles, denoted by [Formula: see text], satisfies [Formula: see text] for [Formula: see text]; and [Formula: see text], [Formula: see text]. It seems that the correct answer for their case [Formula: see text] should be [Formula: see text]. In this paper, we apply Hopf bifurcation theory and normal form computation, and perturb the isolated, nondegenerate center (the origin) to prove that [Formula: see text] for [Formula: see text]; and [Formula: see text] for [Formula: see text], which improve Giné’s results with one more limit cycle for each case.

scholarly journals On the non existence of periodic orbits for a class of two dimensional Kolmogorov systems

2021 β—½ Β 
Vol 2021 (1) β—½ Β 
pp. 1-11
Author(s): Β 
Rachid Boukoucha
Keyword(s): Β 
Periodic Orbits β—½ Β 
Limit Cycles β—½ Β 
Two Dimensional β—½ Β 
Homogeneous Polynomials

Abstract In this paper we characterize the integrability and the non-existence of limit cycles of Kolmogorov systems of the form x β€² = x ( B 1 ( x , y ) ln | A 3 ( x , y ) A 4 ( x , y ) | + B 3 ( x , y ) ln | A 1 ( x , y ) A 2 ( x , y ) | ) , y β€² = y ( B 2 ( x , y ) ln | A 5 ( x , y ) A 6 ( x , y ) | + B 3 ( x , y ) ln | A 1 ( x , y ) A 2 ( x , y ) | ) \matrix{{x' = x\left( {{B_1}\left( {x,y} \right)\ln \left| {{{{A_3}\left( {x,y} \right)} \over {{A_4}\left( {x,y} \right)}}} \right| + {B_3}\left( {x,y} \right)\ln \left| {{{{A_1}\left( {x,y} \right)} \over {{A_2}\left( {x,y} \right)}}} \right|} \right),} \hfill \cr {y' = y\left( {{B_2}\left( {x,y} \right)\ln \left| {{{{A_5}\left( {x,y} \right)} \over {{A_6}\left( {x,y} \right)}}} \right| + {B_3}\left( {x,y} \right)\ln \left| {{{{A_1}\left( {x,y} \right)} \over {{A_2}\left( {x,y} \right)}}} \right|} \right)} \hfill \cr } where A 1 (x, y), A 2 (x, y), A 3 (x, y), A 4 (x, y), A 5 (x, y), A 6 (x, y), B 1 (x, y), B 2 (x, y), B 3 (x, y) are homogeneous polynomials of degree a, a, b, b, c, c, n, n, m respectively. Concrete example exhibiting the applicability of our result is introduced.

Word problems related to periodic solutions of a non-autonomous system

1990 β—½ Β 
Vol 108 (1) β—½ Β 
pp. 127-151 β—½ Β 
Author(s): Β 
James Devlin
Keyword(s): Β 
Periodic Solutions β—½ Β 
Limit Cycles β—½ Β 
Word Problems β—½ Β 
Autonomous System β—½ Β 
Homogeneous Polynomials β—½ Β 
Image Position β—½ Β 
Abstract Formulation

In this paper, we develop an abstract formulation of a problem which arises in the investigation of the number of limit cycles of systems of the formwhere p and q are homogeneous polynomials. This is part of the much wider study of Hilbert's sixteenth problem, in which information is sought about the number of limit cycles of systems of the formwhere P and Q are polynomials, and their possible configurations.

Bifurcation of Limit Cycles from the Center of a Quintic System via the Averaging Method

2017 β—½ Β 
Vol 27 (05) β—½ Β 
pp. 1750072 β—½ Β 
Author(s): Β 
Bo Huang
Keyword(s): Β 
Limit Cycles β—½ Β 
Upper Bound β—½ Β 
Averaging Method β—½ Β 
Unperturbed System β—½ Β 
Homogeneous Polynomials β—½ Β 
Bifurcation Of Limit Cycles β—½ Β 
First Order β—½ Β 
Period Annulus

This paper deals with the bifurcation of limit cycles for a quintic system with one center. Using the averaging method, we explain how limit cycles can bifurcate from the periodic annulus around the center of the considered system by adding perturbed terms which are the sum of homogeneous polynomials of degree [Formula: see text] for [Formula: see text]. We show that up to first-order averaging, at most five limit cycles can bifurcate from the period annulus of the unperturbed system for [Formula: see text], at most [Formula: see text] limit cycles can bifurcate from the periodic annulus of the unperturbed system for any [Formula: see text], and the upper bound is sharp for [Formula: see text] and for [Formula: see text].

Limit cycles of a class of polynomial systems

1988 β—½ Β 
Vol 109 (1-2) β—½ Β 
pp. 187-199 β—½ Β 
Author(s): Β 
Marc Carbonell β—½ Β 
Jaume Llibre
Keyword(s): Β 
Limit Cycles β—½ Β 
Vector Fields β—½ Β 
Polynomial Systems β—½ Β 
Polynomial Vector β—½ Β 
Homogeneous Polynomials β—½ Β 
Polynomial Vector Fields

SynopsisWe study the class of polynomial vector fields of the form = Ξ±x β€” y + Pn(x, y), = x + Ξ±y + Qn(x, y), where Pn and Qn are homogeneous polynomials of degree n. If we define the functions f(x, y) = xPn(x, y) + yQn(x, y) and g(x, y) = xQn(x, y)βˆ’yPn(x, y), we characterise the number of limit cycles for this class when the function g(Ξ±g βˆ’ f) does not change sign.

scholarly journals On the limit cycles of polynomial differential systems with homogeneous nonlinearities

2000 β—½ Β 
Vol 43 (3) β—½ Β 
pp. 529-543 β—½ Β 
Author(s): Β 
Chengzhi Li β—½ Β 
Weigu Li β—½ Β 
Jaume Llibre β—½ Β 
Zhifen Zhang
Keyword(s): Β 
Differential Equations β—½ Β 
Periodic Orbits β—½ Β 
Limit Cycles β—½ Β 
Homogeneous Polynomials β—½ Β 
Differential Systems β—½ Β 
Polynomial Differential Equations β—½ Β 
Polynomial Differential Systems

AbstractWe consider three classes of polynomial differential equations of the form αΊ‹ = y + establish Pn (x, y), ẏ = x + Qn (x, y), where establish Pn and Qn are homogeneous polynomials of degree n, having a non-Hamiltonian centre at the origin. By using a method different from the classical ones, we study the limit cycles that bifurcate from the periodic orbits of such centres when we perturb them inside the class of all polynomial differential systems of the above form. A more detailed study is made for the particular cases of degree n = 2 and n = 3.

scholarly journals On the Limit Cycles of the Polynomial Differential Systems with a Linear Node and Homogeneous Nonlinearities

2014 β—½ Β 
Vol 24 (05) β—½ Β 
pp. 1450065 β—½ Β 
Author(s): Β 
Jaume Llibre β—½ Β 
Jiang Yu β—½ Β 
Xiang Zhang
Keyword(s): Β 
Differential Equations β—½ Β 
Limit Cycles β—½ Β 
Homogeneous Polynomials β—½ Β 
Differential Systems β—½ Β 
Polynomial Differential Equations β—½ Β 
Polynomial Differential Systems β—½ Β 
Existence And Nonexistence

We consider the class of polynomial differential equations αΊ‹ = Ξ»x + Pn(x, y), ẏ = ΞΌy + Qn(x, y) in ℝ2 where Pn(x, y) and Qn(x, y) are homogeneous polynomials of degree n > 1 and Ξ» β‰  ΞΌ, i.e. the class of polynomial differential systems with a linear node with different eigenvalues and homogeneous nonlinearities. For this class of polynomial differential equations, we study the existence and nonexistence of limit cycles surrounding the node localized at the origin of coordinates.

NON-ALGEBRAIC LIMIT CYCLES FOR PARAMETRIZED PLANAR POLYNOMIAL SYSTEMS

2007 β—½ Β 
Vol 18 (02) β—½ Β 
pp. 179-189 β—½ Β 
Author(s): Β 
KHALIL I. T. AL-DOSARY
Keyword(s): Β 
Limit Cycles β—½ Β 
Polynomial Systems β—½ Β 
Homogeneous Polynomials β—½ Β 
The Family β—½ Β 
Planar Systems β—½ Β 
Algebraic Limit Cycles β—½ Β 
Algebraic Limit

In this paper, we determine conditions for planar systems of the form [Formula: see text] where a, b and c are real constants, to possess non-algebraic limit cycles. This is done as an application of a former theorem gives description of the existence of the non-algebraic limit cycles of the family of systems: [Formula: see text] where Pn(x,y), Qn(x,y) and Rm(x,y) are homogeneous polynomials of degrees n, n and m respectively with n < m and n is odd, m is even. The tool for proving these results is based on a method developed in [7].

scholarly journals On the Limit Cycles of Linear Differential Systems with Homogeneous Nonlinearities

2015 β—½ Β 
Vol 58 (4) β—½ Β 
pp. 818-823 β—½ Β 
Author(s): Β 
Jaume Llibre β—½ Β 
Xiang Zhang
Keyword(s): Β 
Limit Cycles β—½ Β 
Homogeneous Polynomials β—½ Β 
Differential Systems β—½ Β 
Polynomial Differential Systems β—½ Β 
Linear Differential Systems β—½ Β 
Nonexistence And Existence β—½ Β 
Linear Differential

AbstractWe consider the class of polynomial differential systems of the form , where Pn and Qn are homogeneous polynomials of degree n. For this class of differential systems we summarize the known results for the existence of limit cycles, and we provide new results for their nonexistence and existence

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