Resonance regions for families of torus maps

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Abstract
A resonance region for a family of torus maps f:Tn to Tn 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. The authors discuss the significance of their findings for the transition to chaos from three-frequency quasiperiodic motion.