It is well known that the popular Störmer—Cowell class of linear multistep methods; for the initial value problem for a second-order ordinary differential system suffers a deficiency, sometimes known as orbital instability, when the stepnumber exceeds two. For a test problem describing uniform motion in a circular orbit, the numerical solution afforded by such a method spirals inwards for all values of the steplength. Various modified Störmer—Cowell methods have been proposed to overcome this deficiency, but they all require a priori knowledge of the frequency. In this paper it is shown that certain linear multistep methods of arbitrary stepnumber possess a periodicity property when the product of the steplength and the angular frequency lies within a certain interval, the interval of periodicity. The symmetry conditions under which a linear multistep method possesses a non-vanishing interval of periodicity are established. The superiority of a symmertic method with non-vanishing interval of periodicity over a comparable Störmer—Cowell method is demonstrated numerically.