Wrinkled tori and bursts due to resonant temporal forcing

The effect of resonant temporal forcing on a system undergoing a Hopf bifurcation with D4 (square) symmetry is studied. The forcing breaks the continuous normal form symmetry of the governing amplitude equations, but the D4 symmetry is preserved. For this system, it is shown that bursts with very large dynamic range may occur. The bursts are associated with visits near solutions “at infinity”, and are related to those found previously for the Hopf bifurcation with broken D4 symmetry [Moehlis and Knobloch, Physica D 135 (2000) 263]. The regime in which an attracting quasiperiodic solution that exists in the absence of forcing “wrinkles” into chaos as the amplitude of the forcing increases is investigated in detail. The overall behavior is governed by the approach of the resulting attractor to solutions at infinity. Windows with stable periodic solutions are found and associated with the traversal in parameter space through Arnol’d tongues. Other aspects of the dynamics are related to the presence of a new type of gluing bifurcation (which we call a “supergluing bifurcation”).