2026-04-17T01:13:57Zhttps://recercat.cat/oai/request

oai:recercat.cat:2117/8472025-07-16T23:43:18Zcom_2072_1033col_2072_452950

The period function for second-order quadratic ODEs is monotone Gasull Embid, Armengol Guillamon Grabolosa, Antoni Villadelprat Yagüe, Jordi Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions Differential equations Differentiable dynamical systems period function Equacions diferencials ordinàries Sistemes dinàmics diferenciables Classificació AMS::34 Ordinary differential equations::34C Qualitative theory Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory Very little is known about the period function for large families of centers. In one of the pioneering works on this problem, Chicone [?] conjectured that all the centers encountered in the family of second-order differential equations ¨x = V (x, ˙ x), being V a quadratic polynomial, should have a monotone period function. Chicone solved some of the cases but some others remain still unsolved. In this paper we fill up these gaps by using a new technique based on the existence of Lie symmetries and presented in [?]. This technique can be used as well to reprove all the cases that were already solved, providing in this way a compact proof for all the quadratic second-order differential equations. We also prove that this property on the period function is no longer true when V is a polynomial which nonlinear part is homogeneous of degree n > 2. 2003 Article https://hdl.handle.net/2117/847 eng http://creativecommons.org/licenses/by-nc-nd/2.5/es/ Open Access Attribution-NonCommercial-NoDerivs 2.5 Spain 21 application/pdf