Coupling of simple square abstract oscillators: synchronized periodic solutions are generic

Abstract oscillators are simple differential equations on the lift of a torus. In particular, square abstract oscillators have interesting computational properties. The study of the synchronization of two such oscillators as there relative speed is varied is shown to be very related to the study of one of them forced by an external periodic signal. The computation of the bounds of the synchronization domains in some parameter space, sometime known as Arnold’s tongues, is formally possible. It is shown that the tongues can be deduced from two particular ones by a sequence of transformations associated with a kind of continued fraction expansion of the winding number. So it is possible to find formal expressions for the bounds of the tongues from which it is possible to compute the measure of the set of forcing periods for which the solution is not periodic. It is shown that, as long as the forcing is not null, this measure is null, a counter intuitive result. For the coupled case it means that as soon as the coupling is dissymmetric the oscillators are almost always synchronized giving a periodic solution.