Abstract
A map on a torus is called “quasiperiodic” if there is a change of variables which converts it into a pure rotation in each coordinate of the torus. We develop a numerical method for finding this change of variables, a method that can be used effectively to determine how smooth (i.e., differentiable) the change of variables is, even in cases with large nonlinearities. Our method relies on fast and accurate estimates of limits of ergodic averages. Instead of uniform averages that assign equal weights to points along the trajectory of N points, we consider averages with a non-uniform distribution of weights, weighing the early and late points of the trajectory much less than those near the midpoint N∕2. We provide a one-dimensional quasiperiodic map as an example and show that our weighted averages converge far faster than the usual rate of O(1∕N), provided f is sufficiently differentiable. We use this method to efficiently numerically compute rotation numbers, invariant densities, conjugacies of quasiperiodic systems, and to provide evidence that the changes of variables are (real) analytic.
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Das, S., Saiki, Y., Sander, E., Yorke, J.A. (2016). Quasiperiodicity: Rotation Numbers. In: Skiadas, C. (eds) The Foundations of Chaos Revisited: From Poincaré to Recent Advancements. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-29701-9_7
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DOI: https://doi.org/10.1007/978-3-319-29701-9_7
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