We study the transition to chaos caused by overlap of resonances in a dissipative standardlike map exhibiting recurrence of invariant circles. In particular, the effects of the recurrence on the structure of a critical line at which the motion is always mode locked are investigated. When there is no recurrence of invariant circles, a smooth critical line exists. In the recurrence case, however, the critical line is folded back and has discontinuous parts. At this line the mode-locked intervals trace a folded devil’s staircase with isolated parts. Scaling behavior of the complementary set to the folded staircase is also discussed.

• Received 13 November 1991

DOI:https://doi.org/10.1103/PhysRevA.45.5480