The period function of some polynomial systems of arbitrary degree (Q1907728)

The author extends \textit{W. A. Coppel}'s and \textit{L. Gavrilov}'s result [Differ. Integral Equ. 6, No. 6, 1357-1365 (1993; Zbl 0780.34023)] to Hamiltonian polynomial systems in the plane which possess a centre. He considers the case when the Hamiltonian is of the form \(H(x,y) = {1 \over 2} (x^2 + y^2) + K_{n + 1} (x,y)\), \(K_{n + 1}\) is homogeneous polynomial of degree \(n + 1\), \(n\) an even positive integer. He proves that the period function is a strictly increasing function of the energy. He also considers an arbitrary homogeneous vector field which possesses a centre and proves that the period function is a strictly monotone function of the distance of the closed curve to the centre, except when \(n = 1\), in which case the system is isochronous.