Kirthana Rajasekar: Synchronization in quasi-periodically forced circle diffeomorphisms

Abstract: In 1984, Grebogi et al. showed the occurrence of strange non-chaotic attractors in a specific class of quasi-periodically forced systems. Since then, various numerical studies have indicated the prevalence of geometrically intricate invariant graphs in quasi-periodically forced systems. This has continued to motivate theoretical results that can uncover the underlying mechanisms that generate such invariant structures.

In this talk, we discuss quasi-periodically forced circle diffeomorphisms, with a focus on circle diffeomorphisms which have exactly two attracting and two repelling fixed points. Under a sufficiently weak forcing, one can prove that the corresponding skew-product map is uniformly hyperbolic. Under a strong quasi-periodic forcing, the dynamics become notably more complicated. In such cases, for certain weak conditions of asymmetry on the fixed points, we establish that the corresponding map \(F_\omega\) exhibits synchronisation along the fibre direction for a set of frequencies \(\omega\) of positive measure. Further, we show that the hyperbolic behaviour in the system can depend sensitively on the arithmetic properties of the frequency.