Quasiperiodic approximation of nonlinear phenomenon approximating complex, chaotic systems via torus rotations

A dynamical system comprises of a collection Ω of states, and a transition law f : Ω -> Ω. Th role of f is to scatter the states all around Ω, with each successive iteration. The goal of dynamical systems theory is to track how points stay together or diverge. The class of dynamics that are most notorious for being intractable are chaotic systems. 

It has been observed that even the most dissipative or chaotic systems have periodic structures embedded in them. This has been proved in mutiple ways, using the technique of towers [5], differential geometry [2, 3, 4], and operator theory [1]. Recent numerical investigations into real world data has revealed that data often appears to be driven by a primarily periodic source, along with some chaotic perturbations [6,7]. All of this suggest a strong possibility of presenting arbitrary dynamical systems to be predominantly qusi-periodic.

Future directions. 

1) Implement the algorithms from [1] in Julia, to extract almost-periodic patterns in timeseries generated by data.

2) Model the switching between dominant Koopman modes as a finite state Markov transition.

3) Establish an universal approximation theorem for dynamical systems, using Quasiperiodically driven dynamical systems as the medium.

Prerequisites. 

1) Programming : familiarity with Matlab / Julia; familiarity with programming techniques in Linear Algebra

2) Linear Algebra : matrix theory, SVD, eigen-decomposition, least squares methods

3) Analysis : continuity and differntiability, limits and convergence

4) Functional analysis : Banach spaces, Hilbert spaces, kernel integrals

5) Measure theory : probability measures, absolute continuity, Markov transitions, Borel sets, abstract Measure theory