Stochastic perturbations and Ulam's method for W-shaped maps

Paweł Góra,Abraham Boyarsky

doi:10.3934/dcds.2013.33.1937

Abstract

For a discrete dynamical system given by a map $\tau :I\rightarrow I$, thelong term behavior is described by the probability density function (pdf) ofan absolutely continuous invariant measure. This pdf is the fixed point ofthe Frobenius-Perron operator on $L^{1}(I)$ induced by $\tau$. Ulamsuggested a numerical procedure for approximating a pdf by using matrixapproximations to the Frobenius-Perron operator. In [12] Li provedthe convergence for maps which are piecewise $C^{2}$ and satisfy$|\tau'| >2.$ In this paper we will consider a largerclass of maps with weaker smoothness conditions and a harmonic slopecondition which permits slopes equal to $\pm $2. Using a generalizedLasota-Yorke inequality [4], we establish convergence for the Ulamapproximation method for this larger class of maps. Ulam's methodis a special case of small stochastic perturbations. We obtain stability of the pdf under such perturbations.Although our conditions apply to manymaps, there are important examples which do not satisfy these conditions,for example the $W$-map [7]. The $W$-map is highly unstable in the sense thatit is possible to construct perturbations $W_a$ withabsolutely continuous invariant measures (acim) $\mu_a$such that $\mu_a$ converge to a singular measure although $W_a$ converge to $W$. We prove the convergence of Ulam's methodfor the $W$-map by direct calculations.

Highlights

The W -map is highly unstable in the sense that it is possible to construct perturbations Wa with absolutely continuous invariant measures μa such that μa converge to a singular measure Wa converge to W
The long term behavior of a discrete time chaotic dynamical system τ : I → I is described by the probability density function of an absolutely continuous invariant measure
In [10] we show that acims of maps close to the W map converge to a singular measure rather than the pdf for the W map

Summary

Introduction

The long term behavior of a discrete time chaotic dynamical system τ : I → I is described by the probability density function (pdf) of an absolutely continuous invariant measure (acim). The pdf is the fixed point of the Frobenius-Perron operator on L1(I) induced by τ, but only in very simple cases can it be computed analytically. The most used and best understood is Ulam’s method [15] which approximates the invariant pdf by invariant vectors of finite dimensional matrices which are restrictions of the Frobenius-Perron operator to subspaces of L1(I) generated by finite partitions of I. The convergence of Ulam’s method has been proved in [12], for piecewise C2, piecewise expanding maps of interval satisfying. Piecewise expanding maps of an interval, absolutely continuous invariant measures, Frobenius-Perron operator, Markov maps, W-shaped maps, Ulam’s method, harmonic average of slopes.

PAWEL GO RA AND ABRAHAM BOYARSKY

If τ satisfies

For i