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COMPUTING THE INVARIANT MEASURE AND THE LYAPUNOV EXPONENT FOR ONE-DIMENSIONAL MAPS USING A MEASURE-PRESERVING POLYNOMIAL BASIS

Aston, PJ and Junge, O (2014) COMPUTING THE INVARIANT MEASURE AND THE LYAPUNOV EXPONENT FOR ONE-DIMENSIONAL MAPS USING A MEASURE-PRESERVING POLYNOMIAL BASIS MATHEMATICS OF COMPUTATION, 83 (288), PII S . pp. 1869-1902.

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Abstract

We consider a generalisation of Ulam's method for approximating invariant densities of one-dimensional 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.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
AuthorsEmailORCID
Aston, PJUNSPECIFIEDUNSPECIFIED
Junge, OUNSPECIFIEDUNSPECIFIED
Date : 1 July 2014
Identification Number : 10.1090/S0025-5718-2013-02811-6
Uncontrolled Keywords : Science & Technology, Physical Sciences, Mathematics, Applied, Mathematics, MATHEMATICS, APPLIED, FROBENIUS-PERRON OPERATORS, FINITE APPROXIMATIONS, ULAMS METHOD, INTERVAL MAPS, DYNAMICS, DISCRETIZATION, CONVERGENCE, ATTRACTORS, CONJECTURE
Related URLs :
Additional Information : © Copyright 2013 American Mathematical Society. First published in MATHEMATICS OF COMPUTATION in volume 83(288), published by the American Mathematical Society.
Depositing User : Symplectic Elements
Date Deposited : 05 Aug 2014 14:07
Last Modified : 17 Jan 2015 14:48
URI: http://epubs.surrey.ac.uk/id/eprint/805803

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