Monotonous Period Function for Equivariant Differential Equations with Homogeneous Nonlinearities

Abstract We prove that the period function of the center at the origin of the $$\mathbb {Z}_k$$ <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 $$\dot{z}=iz+a(z\overline{z})^nz^{k+1}, a\ne 0,$$ <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.