2025/06/24 by Armengol Gasull, David Rojas · 1 voice
Mathematics · #Advanced Differential Equations and Dynamical Systems #Nonlinear Differential Equations Analysis #Numerical methods for differential equations
paper · pdf · doi:10.1007/s00009-025-02879-2
openalex publication_date 2025/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
Abstract We prove that the period function of the center at the origin of the \mathbb Zk <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> -equivariant differential equation z=iz+a(zz)nzk+1, a≠ 0, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>z</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mi>i</mml:mi> <mml:mi>z</mml:mi> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mover> <mml:mi>z</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:msup> <mml:mi>z</mml:mi> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> is monotonous decreasing for all n and k positive integers, solving a conjecture about them. We show this result as corollary of proving that the period function of the center at the origin of a sub-family of the reversible quadratic centers is monotonous decreasing as well.