As a model displaying typical features of two-frequency quasiperiodically forced systems, we discuss the circle map with quasiperiodic coupling. We present numerical and analytical evidence for the existence of strange nonchaotic attractors, and we use examples to illustrate various types of dynamical behavior that can arise in typical quasiperiodically forced systems. We investigate the behavior of the system in the two-dimensional parameter plane of nonlinearity strength versus one of the driving frequencies. We find that the set in this parameter plane for which the system exhibits strange nonchaotic attractors has Cantor-like structure and is enclosed between two critical curves. One of these curves marks the transition from three-frequency quasiperiodic attractors to strange nonchaotic attractors; the other marks the transition from strange nonchaotic attractors to chaotic attractors. This suggests a possible route to chaos in two-frequency quasiperiodically forced systems: (three-frequency quasiperiodicity)→(strange nonchaotic behavior)→(chaos).
Ding, M., Grebogi, C., & Ott, E. (1989). Evolution of attractors in quasiperiodically forced systems: From quasiperiodic to strange nonchaotic to chaotic. Physical Review A, 39(5), 2593-2598. https://doi.org/10.1103/PhysRevA.39.2593