Differential and Integral Equations

The period function of some polynomial systems of arbitrary degree

C. B. Collins

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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