Resonance and weak chaos in quasiperiodically-forced circle maps

In this paper, we distinguish between four categories of dynamics for quasiperiodically-forced (QPF) circle maps: resonant and incommensurate regular dynamics, and strongly and weakly chaotic dynamics, using the weighted Birkhoff average (WBA). Regular orbits can be classified by their rotation vectors, and these can be rapidly computed to machine precision using the WBA. These orbits can be resonant or incommensurate and we distinguish between these by computing their “resonance order,” allowing us to quickly identify and observe the geometric properties of a large set of Arnold tongues. When the dynamics is chaotic the WBA converges slowly. Orbits that are not regular can be strongly chaotic, when they have a positive Lyapunov exponent, or weakly chaotic when the maximal Lyapunov exponent is not positive. The latter correspond to the strange nonchaotic attractors (SNA) that have been observed in QPF circle maps beginning with models introduced by Ding, Grebogi, and Ott. The WBA provides an efficient new technique to find SNAs, and allows us to accurately compute the proportions of each of the four orbit types as a function of map parameters.