Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems

Abstract

In this paper we completely characterize trivial polynomial Hamiltonian isochronous centers of degrees $5$ and $7$. Precisely, we provide simple formulas, up to linear change of coordinates, for the Hamiltonians of the form $H = \left(f_1^2 + f_2^2 \right)/2$, where $f = (f_1, f_2): \mathbb{R}^2\to \mathbb{R}^2$ is a polynomial map with $\det D f = 1$, $f(0,0) = (0,0)$ and the degree of $f$ is $3$ or $4$.