Abstract Consider an equation of the form x εh(x, x) x 0 where ε is small. Such an equation is in a sense close to the simple harmonic equation x x 0, whose phase diagram consists of circles centred on the origin. It should be possible to take advantage of this fact to construct approximate solutions: the phase paths will be nearly circular for ε small enough. However, the original equation will not, in general, have a centre at the origin. The approximate and the exact solutions may differ only by a little over a single cycle, but the difference may prevent the paths from closing; apart from exceptional paths, which are limit cycles. The phase diagram will generally consist of slowly changing spirals and, possibly, limit cycles, all being close to circular. We show several methods for estimating the radii of limit cycles and for detecting a centre.