Centers of cubic polynomial differential systems

Gerardo H. Anacona; Bruno R. Freitas; Jaume Llibre

doi:10.3934/cpaa.2025072

Article Contents

2025, Volume 24, Issue 11: 2130-2145. Doi: 10.3934/cpaa.2025072

This issue Previous Article Homoclinic and ground state solutions of partial difference equations with parametric potentials Next Article Multiplicity of positive solutions to linearly coupled Schrödinger systems involving critical exponent

Centers of cubic polynomial differential systems

1.
Institute of Mathematics and Statistics of Federal University of Goiás, Avenida Esperança s/n, Campus Samambaia, 74690-900, Goiânia, Goiás, Brazil
2.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

^*Corresponding author: Bruno R. Freitas
^*Corresponding author: Bruno R. Freitas

Received: February 2025

Early access: May 26, 2025

Published online: May 26, 2025

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Abstract

An equilibrium point $ p $ of a differential system in the plane $ {\mathbb{R}}^2 $ is a center if there exists a neighborhood $ U $ of $ p $ such that $ U\setminus \{p\} $ is filled with periodic orbits. A difficult classical problem in the qualitative theory of differential systems in the plane is the problem of distinguishing between a focus and a center. In this paper we characterize when the origin of coordinates is a center of the following cubic polynomial differential systems

$ \begin{equation*} \dot{x} = -y, \qquad \dot{y} = x + a_{1} x^{2}+ a_{2} x y+ a_{3} y^{2}+ A, \end{equation*} $

where $ A $ is an arbitrary nonzero monomial of degree 3. Moreover we provide all topologically different phase portraits when $ A = a_{4}x^{3} $.

Keywords:

Mathematics Subject Classification: Primary: 34A34, 34C07, 37C27; Secondary: 34C25, 34C14.

Citation:

Full Text(HTML)

Figure 1. $ a_{2} = 0 $. In figures $ (k)-(o) $ we have $ a_{1}<0 $ and $ 0<a_{4}<\frac{a_{1}^{2}}{4} $. In figures $ (p)-(t) $ we have $ a_{1}>0 $ and $ 0<a_{4}<\frac{a_{1}^{2}}{4} $

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Figure 2. $ a_{1} = a_{3} = 0 $

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Figure 3. The Poincaré sphere, the point $ p $ in the tangent plane to the point $ (0,0,1) $ of the sphere has two projetions on the sphere $ p_{1} $ and $ p_{2} $

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Figure 4. Blow up at $ (0,0) $ when $ a_{1}^{2}<4 a_{4} $. Case $ a_{3} = a_{4}>0 $

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Figure 5. Possible phase portrait of system (6) when the origin is the unique finite equilibrium and $ a_{3} = a_{4}>0 $

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Figure 6. Possible phase portrait of system (6) when the origin is the unique finite equilibrium and $ a_{3} = 0 $ and $ a_{1}^{2}<4a_{4} $

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Figure 7. Possible phase portrait of system (6) when $ a_{3} = 0 $, $ a_{1}^{2} = 4a_{4} $

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Figure 8. Possible phase portrait of system (6) when $ a_{3} = 0 $

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Figure 9. Possible phase portrait of system (6) when $ a_{3}<0 $

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Figure 10. Blow up at $ (0,0) $ when $ a_{2}^{2}<8 a_{4} $

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