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URI Permanente para esta coleçãohttps://repositorio.univasf.edu.br/handle/123456789/1407
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Item Conjugação analítica entre sistemas reversíveis e hamiltonianos no plano(2026-07-08) Nacimento, Francisco José dos Santos; https://orcid.org/0009-0004-9536-7979; http://lattes.cnpq.br/9091581147583991This paper investigates whether planar analytic reversible vector fields in a neighborhood of a nondegenerate equilibrium point can be locally described by analytic Hamiltonian systems. We prove that every planar analytic vector field reversible with re- spect to the linear involution $𝑅(𝑥, 𝑦) = (𝑥, −𝑦)$ and possessing a nondegenerate equilibrium point is locally analytically con- jugate to a planar Hamiltonian system. This result promotes to the analytic setting an equivalence that was previously known only at the formal level.Item Mechanical Normal Forms for Analytic Centers(2026-02-28) Nacimento, Francisco José dos Santos; https://orcid.org/0009-0004-9536-7979; http://lattes.cnpq.br/9091581147583991Item Conjugação analítica entre sistemas diferenciais planares e sistemas potenciais(2024-12-19) Nacimento, Francisco José dos Santos; https://orcid.org/0009-0004-9536-7979; http://lattes.cnpq.br/9091581147583991The classic Poincaré Normal Form Theorem states that a critical point of an analytic planar vector field is a non-degenerate center if and only if there is an analytic coordinate change such that in the new coordinates the vector field initial is of the form $f(x^2+y^2)\big(y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}\big)$, where $f$ is an analytic function defined in a neighborhood of the origin such that $f(0)>0.$ In this article it is proved that an analytical planar vector field with a non-degenerate center at $(0,0)$ is analytically conjugate, in a neighborhood of $(0,0)$, to a Hamiltonian vector field of the form $y\frac{\partial}{\partial x}-V'(x)\frac{\partial}{\partial y}$, where $V$ is an analytic function defined in a neighborhood of the origin such that $V(0)=V'(0)=0$ and $V''(0)>0.$ This result is a partial answer to a problem proposed by Chicone in 1987.Item Global normalizations for centers of planar vector fields(2025-01-15) Grotta-Ragazzo , C.; Nacimento, Francisco José dos Santos ; http://lattes.cnpq.br/9091581147583991