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The period function of some polynomial systems of arbitrary degree. (English) Zbl 0849.34025

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)

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

Citations:

Zbl 0780.34023
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