Center problem for generalized Λ-Ω differential systems
The Λ-Ω differential systems are the real planar polynomial differential equations of degree m of the form ˙ x = -y(1 + Λ) + xΩ, ˙ y = x(1 + Λ) + yΩ, where Λ = Λ(x,y) and Ω = Ω(x,y) are polynomials of degree at most m-1 such that Λ(0,0) = Ω(0,0) = 0. We study the center problem for these Λ-Ω systems...
| Autores: | , , |
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| Formato: | artículo |
| Fecha de publicación: | 2018 |
| País: | España |
| Recursos: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:222607 |
| Acesso em linha: | https://ddd.uab.cat/record/222607 |
| Access Level: | acceso abierto |
| Palavra-chave: | Darboux first integral Linear type centers Poincaré-Liapunov Theorem Reeb integrating factor Weak center |
| Resumo: | The Λ-Ω differential systems are the real planar polynomial differential equations of degree m of the form ˙ x = -y(1 + Λ) + xΩ, ˙ y = x(1 + Λ) + yΩ, where Λ = Λ(x,y) and Ω = Ω(x,y) are polynomials of degree at most m-1 such that Λ(0,0) = Ω(0,0) = 0. We study the center problem for these Λ-Ω systems. A planar vector field with linear type center can be written as an Λ-Ω system if and only if the Poincaré-Liapunov first integral is of the form F = 1 2 (x2 + y2)(1 + O(x,y)). The main objective of this paper is to study the center problem for Λ-Ω systems of degree m with Λ = µ(a2x - a1y), and Ω = a1x + a2y + m-1 ∑ j=2 Ωj, where µ, a1, a2 are constants and Ωj = Ωj(x,y) is a homogenous polynomial of degree j, for j = 2,...,m-1. We prove the following results. Assuming that m = 2,3,4,5 and (µ + (m-2))(a2 1 + a2 2) ̸= 0 and m-2 ∑ j=2 Ωj ̸= 0 then the Λ-Ω system has a weak center at the origin if and only if these systems after a linear change of variables (x,y) -→ (X,Y ) are invariant under the transformations (X,Y,t) -→ (-X,Y,-t). If (µ + (m-2))(a2 1 + a2 2) = 0 and m-2 ∑ j=1 Ωj = 0 then the origin is a weak center. We observe that the main difficulty to prove this result for m. |
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