Multifractality, multifractal phase transitions, and symmetry-increasing bifurcations in ac-driven phase-slip centers

Chaos at and beyond onset is studied for nonequilibrium current-carrying dissipative states in quasi-one-dimensional dirty superconductors. For the case of ac-driving currents, phase-slip center solutions of the generalized time-dependent Ginzburg-Landau equations show a universal transition at the onset of chaos. For currents below the onset, the pervasive feature is the nonhyperbolicity: homoclinic tangency between stable and unstable manifolds of unstable periodic orbits. Pointwise dimensions α(x) evaluated on the attractors show abnormally low values, indicating regions made of an overlapping of stable and unstable manifolds of saddle orbits. These regions cause a break of self-similarity and a phaselike transition in the multifractal probability measure of the attractor. Finally, one of these tangencies causes a symmetry-increasing bifurcation.