Differential and Integral Equations

The period function of some polynomial systems of arbitrary degree

C. B. Collins

Full-text: Open access


We consider certain vector fields in the plane which possess a centre. The main result is that for Hamiltonian polynomial systems which are of even degree, which possess homogeneous nonlinearities, and which have a centre located at the origin, the period function is a strictly increasing function of the energy, throughout its interval of definition. It is also shown that for nonlinear homogeneous Hamiltonian polynomial vector fields of arbitrary degree which possess a centre, the period function is a strictly decreasing function of the energy. With appropriate modifications, this result is extended to arbitrary homogeneous vector fields which possess a centre, irrespective of their being Hamiltonian or polynomial; the period function is then strictly monotonic, except when the degree of homogeneity is one, when the systems are isochronous.

Article information

Differential Integral Equations, Volume 9, Number 2 (1996), 251-266.

First available in Project Euclid: 3 May 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34C05: Location of integral curves, singular points, limit cycles
Secondary: 58F21 70H05: Hamilton's equations


Collins, C. B. The period function of some polynomial systems of arbitrary degree. Differential Integral Equations 9 (1996), no. 2, 251--266. https://projecteuclid.org/euclid.die/1367603345

Export citation