Collins, C. B. The period function of some polynomial systems of arbitrary degree. (English) Zbl 0849.34025 Differ. Integral Equ. 9, No. 2, 251-266 (1996). The author extends W. A. Coppel’s and 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. Reviewer: A.Reinfelds (Riga) Cited in 8 Documents MSC: 34C25 Periodic solutions to ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems Keywords:Hamiltonian polynomial systems in the plane; period function Citations:Zbl 0780.34023 PDFBibTeX XMLCite \textit{C. B. Collins}, Differ. Integral Equ. 9, No. 2, 251--266 (1996; Zbl 0849.34025)