Title: Cubic planar differential systems with non-algebraic limit cycles enclosing a focus

Authors: Meryem Belattar; Rachid Cheurfa; Ahmed Bendjeddou

Addresses: Laboratory of Applied Mathematics, Department of Mathematics, Faculty of Sciences, Ferhat Abbas University-Sétif 1, P.O. Box 19000, Sétif, Algeria ' Laboratory of Applied Mathematics, Department of Mathematics, Faculty of Sciences, Ferhat Abbas University-Sétif 1, P.O. Box 19000, Sétif, Algeria ' Laboratory of Applied Mathematics, Department of Mathematics, Faculty of Sciences, Ferhat Abbas University-Sétif 1, P.O. Box 19000, Sétif, Algeria

Abstract: In this paper, we consider a class of two-dimensional polynomial differential systems of degree three and we classify all its possible local phase portraits. This shows that at most one limit cycle enclosing a focus at the origin arises. Then, we prove that this system is integrable and in accordance with the qualitative study done before, an explicit non-algebraic limit cycle is obtained. Surprisingly, there is a special case for which this limit cycle becomes algebraic and the system can be fully solved.

Keywords: algebraic limit cycle; Bernoulli equation; first integral; phase portrait; Poincaré return map; transcendental limit cycle.

DOI: 10.1504/IJDSDE.2023.135022

International Journal of Dynamical Systems and Differential Equations, 2023 Vol.13 No.3, pp.197 - 208

Received: 19 Jul 2022
Accepted: 10 Jul 2023
Published online: 27 Nov 2023 *

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