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 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 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.

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

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

In the two articles in Appl. Math. Comput., J. Giné [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 ◽  
Vol 2021 (1) ◽  
pp. 1-11
Author(s):  
Rachid Boukoucha
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Two Dimensional ◽  
Homogeneous Polynomials

Abstract In this paper we characterize the integrability and the non-existence of limit cycles of Kolmogorov systems of the form x ′ = 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 ′ = 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 ◽  
Vol 108 (1) ◽  
pp. 127-151 ◽  
Author(s):  
James Devlin
Keyword(s):  
Periodic Solutions ◽  
Limit Cycles ◽  
Word Problems ◽  
Autonomous System ◽  
Homogeneous Polynomials ◽  
Image Position ◽  
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.

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