Lyapunov exponent and natural invariant density determination of chaotic maps: An iterative maximum entropy ansatz

TL;DR

This paper introduces a novel iterative maximum entropy method to accurately estimate the invariant density and Lyapunov exponent of one-dimensional chaotic maps using finite moments, demonstrating high accuracy and stability.

Contribution

It presents a new function reconstruction technique based on the Hausdorff moment problem and Shannon entropy to determine invariant densities and Lyapunov exponents of nonlinear maps.

Findings

Accurately estimates invariant density and Lyapunov exponent for known chaotic maps.

Demonstrates stability and high accuracy of the method through comparisons.

Successfully applies the method to complex maps without known analytical solutions.

Abstract

We apply the maximum entropy principle to construct the natural invariant density and Lyapunov exponent of one-dimensional chaotic maps. Using a novel function reconstruction technique that is based on the solution of Hausdorff moment problem via maximizing Shannon entropy, we estimate the invariant density and the Lyapunov exponent of nonlinear maps in one-dimension from a knowledge of finite number of moments. The accuracy and the stability of the algorithm are illustrated by comparing our results to a number of nonlinear maps for which the exact analytical results are available. Furthermore, we also consider a very complex example for which no exact analytical result for invariant density is available. A comparison of our results to those available in the literature is also discussed.