SRB - COGEO - Geologia
URI Permanente desta comunidadehttps://repositorio.univasf.edu.br/handle/123456789/1406
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Item Conjugação analítica entre sistemas diferenciais planares e sistemas potenciais(2024-12-19) Nascimento, 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.; Nascimento, Francisco José dos Santos ; http://lattes.cnpq.br/9091581147583991Item Mechanical Normal Forms for Analytic Centers(2026-02-28) Nascimento, Francisco José dos Santos; https://orcid.org/0009-0004-9536-7979; http://lattes.cnpq.br/9091581147583991Item Conjugação analítica entre sistemas reversíveis e hamiltonianos no plano(2026-07-08) Nascimento, 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 Monotonicity of the Period Function in Planar Separable Hamiltonian Systems: An Extension of Chicone’s Criterion(2026-09-17) Nascimento, F.J.S.; https://orcid.org/0009-0004-9536-7979; http://lattes.cnpq.br/9091581147583991