Strange non-chaotic attractors in quasiperiodically forced circle maps

Abstract

The occurrence of strange non-chaotic attractors (SNA) in quasiperiodically forced systems has attracted considerable interest over the last two decades, in particular since it provides a rich class of examples for the possibility of complicated dynamics in the absence of chaos. Their existence was discovered in the early 1980's, independently by Herman for quasiperiodic SL(2,R)-cocycles and by Grebogi et al for so-called 'pinched skew products'. However, except for these two particular classes there are still hardly any rigorous results on the topic, despite a large number of numerical studies which all confirmed the widespread existence of SNA in quasiperiodically forced systems. Here, we prove the existence of SNA in quasiperiodically forced circle maps under rather general conditions, which can be stated in terms of C 1 -estimates. As a consequence, we obtain the existence of strange non-chaotic attractors for parameter sets of positive measure in suitable parameter families. Further, we show that the considered systems have minimal dynamics. The results apply in particular to a forced version of the Arnold circle map. For this particular example, we also describe how the first Arnold tongue collapses and looses its regularity due to the presence of strange non-chaotic attractors and a related unbounded mean motion property.