Resonance and Weak Chaos in Quasiperiodically-Forced Circle Maps

Abstract

In this paper, we focus on a numerical technique, the weighted Birkhoff average (WBA) to distinguish between four categories of dynamics for quasiperiodically-forced circle maps. Regular dynamics can be classified by rotation vectors, and these can be rapidly computed to machine precision using the WBA. Regular orbits can be resonant or incommensurate and we distinguish between these by computing their "resonance order." When the dynamics is chaotic the WBA converges slowly. Such orbits can be strongly chaotic, when they have a positive Lyapunov exponent or weakly chaotic, when the maximal Lyapunov exponent is zero. The latter correspond to the strange nonchaotic attractors (SNA) that have been observed in quasiperiodically-forced circle maps beginning with the models introduced by Ding, Grebogi, and Ott. The WBA provides a 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.

Keywords: Circle maps, Quasiperiodic forcing, Arnold tongues, Resonance, Birkhoff averages, Strange Nonchaotic Attractors

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• Computing Lyapunov Exponents using Weighted Birkhoff Averages
• Weighted Birkhoff averages: Deterministic and probabilistic perspectives