| Abstrakt: |
A trajectory θn := Fn(θ0), n = 0,1,2,... is quasiperiodic if the trajectory lies on and is dense in some d-dimensional torus Td, and there is a choice of coordinates on Td for which F has the form F(θ) = θ + ρ mod 1 for all θ ∈ Td and for some ρ ∈ Td. (For d>1 we always interpret mod1 as being applied to each coordinate.) There is an ancient literature on computing the three rotation rates for the Moon. However, for d > 1, the choice of coordinates that yields the form F(θ) = θ + ρ mod 1 is far from unique and the different choices yield a huge choice of coordinatizations (ρ1,⋯,ρd) of ρ, and these coordinations are dense in Td. Therefore instead one defines the rotation rate ρϕ (also called rotation rate) from the perspective of a map ϕ: Td → S1. This is in effect the approach taken by the Babylonians and we refer to this approach as the "Babylonian Problem": determining the rotation rate ρϕ of the image of a torus trajectory - when the torus trajectory is projected onto a circle, i.e., determining ρϕ from knowledge of ϕ(Fn(θ)). Of course ρϕ depends on ϕ but does not depend on a choice of coordinates for Td. However, even in the case d=1 there has been no general method for computing ρϕ given only the sequence ϕ(θn), though there is a literature dealing with special cases. Here we present our Embedding continuation method for general d for computing ρϕ from the image ϕ(θn) of a trajectory, and show examples for d = 1 and 2. The method is based on the Takens Embedding Theorem and the Birkhoff Ergodic Theorem. [ABSTRACT FROM AUTHOR] |
