Monotonous period function for equivariant differential equations with homogeneous nonlinearities

TL;DR

This paper proves that the period function of certain symmetric differential equations decreases monotonically, confirming a conjecture and extending results to a subclass of reversible quadratic centers.

Contribution

It establishes the monotonicity of the period function for a class of equivariant differential equations, solving an existing conjecture and linking to reversible quadratic centers.

Findings

Period function is monotonically decreasing for all positive integers n and k.

Confirms a conjecture about the monotonicity of the period function.

Extends the result to a subclass of reversible quadratic centers.

Abstract

We prove that the period function of the center at the origin of the $\mathbb{Z}_k$-equivariant differential equation $\dot{z}=iz+a(z\overline{z})^nz^{k+1}, a\ne0,$ 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.