Computing the Invariant Measure and the Lyapunov Exponent for One-Dimensional Maps using a Measure-Preserving Polynomial Basis

TL;DR

This paper introduces a novel polynomial-based method for approximating invariant densities and Lyapunov exponents of one-dimensional chaotic maps, demonstrating improved convergence and efficiency over traditional approaches.

Contribution

It develops a measure-preserving polynomial basis for density approximation and proves enhanced convergence rates for Lyapunov exponent computation.

Findings

The polynomial method achieves higher accuracy in density approximation.

Lyapunov exponent estimates converge one order faster than density estimates.

The approach is efficient for highly accurate Lyapunov exponent calculations.

Abstract

We consider a generalisation of Ulam's method for approximating invariant densities of one-dimensional chaotic maps. Rather than use piecewise constant polynomials to approximate the density, we use polynomials of degree n which are defined by the requirement that they preserve the measure on n+1 neighbouring subintervals. Over the whole interval, this results in a discontinuous piecewise polynomial approximation to the density. We prove error results where this approach is used to approximate smooth densities. We also consider the computation of the Lyapunov exponent using the polynomial density and show that the order of convergence is one order better than for the density itself. Together with using cubic polynomials in the density approximation, this yields a very efficient method for computing highly accurate estimates of the Lyapunov exponent. We illustrate the theoretical findings with some examples.