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Phase-locking for maps of a torus: a computer assisted study.

In the description of bifurcations in a family of maps of an n-torus it is natural to consider phase-locked regions in the parameter space that correspond approximately to the sets of parameter values for which the maps have invariant tori. The extreme case of phase-locking is resonance, where the torus map has a periodic orbit. We study a family of maps of an n-torus that only differ from a family of torus translations by a small nonlinear perturbation. The widths of the phase-locked regions for this family generally increase linearly with the perturbation amplitude. However, this growth varies to a higher power law for families of maps that are given by trigonometric polynomials (the so-called Mathieu-type maps). The exponent of the asymptotic power law can be found by simple arithmetic calculations that relate the spectrum of the trigonometric polynomial to the unperturbed translation. Perturbation theory and these calculations predict that typical resonance regions for the family of Mathieu-type maps are narrow elliptical annuli. All these results are illustrated in a number of numerical examples.