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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
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.
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
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
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