We consider conditions under which the second-order differential equationẍ+f(x)ẋ+g(x)=0has a family of periodic orbits with constant period. This condition is equivalent to seeking conditions under which the two-dimensional autonomous systemẋ=y,ẏ=−g(x)−f(x)yhas a center with constant period: i.e., an isochronous center. In turn, this is equivalent to the latter system being locally linearizable.

In [6], we considered the equationwhere z ∈ ℂ and the pi are real-valued functions; abstract word-problem concepts and techniques were applied to the local problem of the bifurcation of periodic solutions out of the solution Z ≡ 0. This paper is a sequel to [6]; we present an extension of certain concepts given in that paper, and give a global version of some of our word-problem results.

In this paper, we develop an abstract formulation of a problem which arises in the investigation of the number of limit cycles of systems of the formwhere p and q are homogeneous polynomials. This is part of the much wider study of Hilbert's sixteenth problem, in which information is sought about the number of limit cycles of systems of the formwhere P and Q are polynomials, and their possible configurations.