Low noise Markov approximation  reconstructing dynamic phenomenon from data

A dynamical system comprises a collection of states, and a transition law transforms states from one to another. Almost every natural dynamic phenomenon, engineering systems as well as computer algorithms can be presented as a dynamical system. When a dynamical system is measured it generates data. 

The art of data-driven discovery of dynamics is to uncover the underlying dynamics from a stream of data. One can try to simply approximate the transition law from data, but as shown in  [3] this is inadequate for preserving important features of the dynamics such as asymptotic behavior and topology of invariant sets. An alternative was proposed in [1] where it was proposed to add a small randomness to the dynamial law. The result is a Markov transition process with low noise, close approximation of the invariant set as well as the dynamics law.

Future directions. 

1) Implementing codes in Julia language.

2) Finding efficient ways to partition a chaotic attractor

3) Devising a numerical method for constructing Rokhlin towers.

4) Creating a numerical procedure for converting the Markov approximation into a switched flow [2].

5) Finding sufficient conditions for stochastic stability.

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