ABSTRACT

The regular structures of a generic 4 d symplectic map with a mixed phase space are organized by one-parameter families of elliptic 1 d -tori. Such families show prominent bends, gaps, and new branches. We explain these features in terms of bifurcations of the families when crossing a resonance. For these bifurcations, no external parameter has to be varied. Instead, the longitudinal frequency, which varies along the family, plays the role of the bifurcation parameter. As an example, we study two coupled standard maps by visualizing the elliptic and hyperbolic 1 d -tori in a 3 d phase-space slice, local 2 d projections, and frequency space. The observed bifurcations are consistent with the analytical predictions previously obtained for quasi-periodically forced oscillators. Moreover, the new families emerging from such a bifurcation form the skeleton of the corresponding resonance channel.