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Differential Equations Seminar

Evelyn Sander, George Mason University

Monday, October 24, 2016
11:00 AM - 12:00 PM
Mathematics/Psychology : 401
Title: Can rotation numbers be computed from a quasiperiodic trajectory?

Abstract: A trajectory is quasiperiodic if the trajectory lies on and is dense in some d-dimensional torus, and there is a choice of coordinates on the torus for which F has the form F(t) = t + rho (mod 1) for all points in the torus, and for some rho in the torus. There is an extensive literature on determining the coordinates of the vector rho, called the rotation numbers of F.  However, even in the one-dimensional case there has been no general method for computing the vector rho given only the trajectory (u_n), though there is an extensive literature dealing with special cases. I will present a computational  method called the  Embedding Continuation Method for computing some components of r from a trajectory. It is based on the Takens Embedding Theorem and the Birkhoff Ergodic Theorem. There is however a caveat; the coordinates of the rotation vector depend on the choice of coordinates of the torus. I will give a statement of the various sets of possible rotation numbers that rho can yield. I will illustrate these ideas with one- and two-dimensional examples. This talk will be at a level accessible to students.