Limit cycles bifurcated from some reversible quadratic centres with a non-algebraic first integral

Nonlinearity ◽  
2012 ◽  
Vol 25 (6) ◽  
pp. 1653-1660 ◽  
Author(s):  
Changjian Liu
Keyword(s):  
Limit Cycles ◽  
First Integral

scholarly journals Integrability and Limit Cycles via First Integrals

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1736
Author(s):  
Jaume Llibre
Keyword(s):  
Phase Portrait ◽  
Differential System ◽  
Limit Cycles ◽  
Applied Mathematics ◽  
First Integral ◽  
First Integrals ◽  
Differential Systems ◽  
Polynomial Differential Systems

In many problems appearing in applied mathematics in the nonlinear ordinary differential systems, as in physics, chemist, economics, etc., if we have a differential system on a manifold of dimension, two of them having a first integral, then its phase portrait is completely determined. While the existence of first integrals for differential systems on manifolds of a dimension higher than two allows to reduce the dimension of the space in as many dimensions as independent first integrals we have. Hence, to know first integrals is important, but the following question appears: Given a differential system, how to know if it has a first integral? The symmetries of many differential systems force the existence of first integrals. This paper has two main objectives. First, we study how to compute first integrals for polynomial differential systems using the so-called Darboux theory of integrability. Furthermore, second, we show how to use the existence of first integrals for finding limit cycles in piecewise differential systems.

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

2021 ◽  
Vol 55 (1 (254)) ◽  
pp. 1-11
Author(s):  
Rachid Boukoucha
Keyword(s):  
Explicit Form ◽  
Limit Cycles ◽  
First Integral ◽  
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.

scholarly journals Polynomial Inverse Integrating Factors, First Integral and Non-Existence of Limit Cycles in the Plane for Quadratic Systems

2017 ◽  
Vol 5 (2) ◽  
pp. 232
Author(s):  
Ahmed M. Hussien
Keyword(s):  
Limit Cycles ◽  
First Integral ◽  
Quadratic Systems ◽  
Integrating Factor ◽  
Inverse Integrating Factor ◽  
Integrating Factors

The main purpose of this paper is to study the existence of polynomial inverse integrating factor and first integral, and non-existence of limit cycles for all systems. Furthermore, we consider some applications.

scholarly journals On deformation of foliations with a center in the projective space

2001 ◽  
Vol 73 (2) ◽  
pp. 191-196 ◽  
Author(s):  
HOSSEIN MOVASATI
Keyword(s):  
Lower Bound ◽  
Projective Space ◽  
Critical Point ◽  
Limit Cycles ◽  
First Integral ◽  
Real Plane ◽  
Real Polynomial ◽  
Fixed Degree ◽  
Polynomial Differential Equations ◽  
Nondegenerate Critical Point

Let <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif"> be a foliation in the projective space of dimension two with a first integral of the type <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img2.gif">, where F and G are two polynomials on an affine coordinate, <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img3.gif"> = <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img4.gif"> and g.c.d.(p, q) = 1. Let z be a nondegenerate critical point of <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img2.gif">, which is a center singularity of <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif">, and <img src="http:/img/fbpe/aabc/v73n2/ft.gif" alt="ft.gif (149 bytes)" align="middle"> be a deformation of <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif"> in the space of foliations of degree deg(<img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif">) such that its unique deformed singularity <img src="http:/img/fbpe/aabc/v73n2/zt.gif" alt="zt.gif (118 bytes)"> near z persists in being a center. We will prove that the foliation <img src="http:/img/fbpe/aabc/v73n2/ft.gif" alt="ft.gif (149 bytes)" align="middle"> has a first integral of the same type of <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif">. Using the arguments of the proof of this result we will give a lower bound for the maximum number of limit cycles of real polynomial differential equations of a fixed degree in the real plane.

scholarly journals The cyclicity of the period annulus of a reversible quadratic system

2021 ◽  
pp. 1-10
Author(s):  
Changjian Liu ◽  
Chengzhi Li ◽  
Jaume Llibre
Keyword(s):  
Limit Cycles ◽  
Mathematical Proof ◽  
First Integral ◽  
Quadratic System ◽  
Quadratic Polynomial ◽  
Unperturbed System ◽  
Computer Assisted ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Period Annulus

We prove that perturbing the periodic annulus of the reversible quadratic polynomial differential system $\dot x=y+ax^2$ , $\dot y=-x$ with a ≠ 0 inside the class of all quadratic polynomial differential systems we can obtain at most two limit cycles, including their multiplicities. Since the first integral of the unperturbed system contains an exponential function, the traditional methods cannot be applied, except in Figuerasa, Tucker and Villadelprat (2013, J. Diff. Equ., 254, 3647–3663) a computer-assisted method was used. In this paper, we provide a method for studying the problem. This is also the first purely mathematical proof of the conjecture formulated by Dumortier and Roussarie (2009, Discrete Contin. Dyn. Syst., 2, 723–781) for q ⩽ 2. The method may be used in other problems.

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