Open Journal of Mathematical Analysis
Vol. 7 (2023), Issue 1, pp. 71 – 82
ISSN: 2616-8111 (Online) 2616-8103 (Print)
DOI: 10.30538/psrp-oma2023.0124
Limit cycles obtained by perturbing a degenerate center
Nabil Rezaiki\(^{1,*}\) and Amel Boulfoul\(^2\)
\(^{1}\) LMA Laboratory , Department of Mathematics, University of Badji Mokhtar, P.O.Box 12, Annaba, 23000, Algeria
\(^{2}\) Department of mathematics, 20 Aout 1955 University, BP26; El Hadaiek 21000, Skikda, Algeria
Copyright © 2023 Nabil Rezaiki and Amel Boulfoul. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: April 07, 2023 – Accepted: June 23, 2023 – Published: June 30, 2023
Abstract
This paper deals with the maximum number of limit cycles bifurcating from the degenerate centre
\[ \dot{x}=-y(3x^2+y^2),\: \dot{y}=x(x^2-y^2), \]
when we perturb it inside a class of all homogeneous polynomial differential systems of degree \(5\). Using averaging theory of second order, we show that, at most, five limit cycles are produced from the periodic orbits surrounding the degenerate centre under quintic perturbation. In addition, we provide six examples that give rise to exactly \(5, 4, 3, 2, 1\) and \(0\) limit cycles.
\[ \dot{x}=-y(3x^2+y^2),\: \dot{y}=x(x^2-y^2), \]
when we perturb it inside a class of all homogeneous polynomial differential systems of degree \(5\). Using averaging theory of second order, we show that, at most, five limit cycles are produced from the periodic orbits surrounding the degenerate centre under quintic perturbation. In addition, we provide six examples that give rise to exactly \(5, 4, 3, 2, 1\) and \(0\) limit cycles.
Keywords:
Limit cycles; averaging theory; polynomial differential systems; degenerate center