A Markov Chain Approximation of a Segment Description of Chaos

TL;DR

This paper investigates a Markov chain approach to approximate chaos and turbulence in dynamical systems, analyzing its effectiveness on various maps and systems, and exploring the behavior of transition matrices as segment length increases.

Contribution

It introduces a Markov chain approximation for segment descriptions of chaos and examines its properties and performance on different dynamical systems.

Findings

Transition matrices tend to become uniform as segment length increases.

Reynolds averaging performs well on fixed point attractors but poorly on strange attractors.

The Markov chain approach provides insights into the structure of chaotic systems.

Abstract

We test a Markov chain approximation to the segment description (Li, 2007) of chaos (and turbulence) on a tent map, the Minea system, the H\'enon map, and the Lorenz system. For the tent map, we compute the probability transition matrix of the Markov chain on the segments for segment time length (iterations) $T = 1, 2, 3, 100$. The matrix has $1, 2, 4$ tents corresponding to $T = 1, 2, 3$; and is almost uniform for $T = 100$. As $T \ra +\infty$, our conjecture is that the matrix will approach a uniform matrix (i.e. every entry is the same). For the simple fixed point attractor in the Minea system, the Reynolds average performs excellently and better than the maximal probability Markov chain and segment linking. But for the strange attractors in the H\'enon map, and the Lorenz system, the Reynolds average performs very poorly and worse than the maximal probability Markov chain and segment linking.