Resonance regions for families of torus maps

A resonance region for a family of torus maps f: T-n -> T-n is the set of parameter values for which there exists a periodic orbit with a given rotation vector. For generic periodic families, resonance regions are projections of multiply connected manifolds. In many cases these are tori. Numerical studies of the case n = 2 illustrate the complicated internal bifurcation structure of the resonance regions. Codimension-two bifurcations and transversal homoclinic orbits are shown to exist. We discuss the significance of our findings for the transition to chaos from three-frequency quasiperiodic