Rotation Vectors for Torus Maps by the Weighted Birkhoff Average

Abstract

In this paper, we focus on distinguishing between the types of dynamical behavior that occur for typical one- and two-dimensional torus maps, in particular without the assumption of invertibility. We use three fast and accurate numerical methods: weighted Birkhoff averages, Farey trees, and resonance orders. The first of these allows us to distinguish between chaotic and regular orbits, as well as to calculate the frequency vectors for the regular case to high precision. The second method allows us to distinguish between the periodic and quasiperiodic orbits, and the third allows us to distinguish among the quasiperiodic orbits to determine the dimension the resulting attracting tori. We first consider the well-studied Arnold circle map, comparing our results to the universal power law of Jensen and Ecke. We next consider quasiperiodically forced circle maps, inspired by models introduced by Ding, Grebogi, and Ott. We use the Birkhoff average to distinguish between "strong" chaos (positive Lyapunov exponents) and "weak" chaos (strange nonchaotic attractors). Finally, we apply our methods to 2D torus maps, building on the work of Grebogi, Ott, and Yorke. We distinguish incommensurate, resonant, periodic, and chaotic orbits and accurately compute the proportions of each as the strength of the nonlinearity grows. We compute generalizations of Arnold tongues corresponding to resonances and to periodic orbits, and we show that chaos typically begins before the map becomes noninvertible. We show that the proportion of nonresonant orbits does not obey a universal power law like that seen in the 1D case.Comment: Keywords: Circle maps, Quasiperiodic forcing, Arnold tongues, Resonance, Birkhoff averages, Strange nonchaotic attractor