Detecting strange attractors in turbulence: Takens’ theorem and the Navier-Stokes equation

James Robinson

Takens’ celebrated time-delay embedding theorem allows the dynamics of a system to be reconstructed from a series of measurements at equally-spaced intervals of time. However, his original proof was only valid for generic dynamical systems evolving on finite-dimensional smooth manifolds. This was improved by Sauer, Yorke and Casdagli to apply to the attractors of finitedimensional systems, but still excludes models of fluid dynamics and hence any application to physical turbulence. In this talk I will give an idea of the proof of a similar theorem for infinite-dimensional dynamical systems (e.g. parabolic partial differential equations). I will also give a simple proof of a result of Yorke that any periodic orbit of an ordinary differential equation ẋ = f(x) with |f(x) − f(y)| ≤ L|x − y| must have period at least 1/L, and a related infinite-dimensional result.