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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Computing the invariant measure and the Lyapunov exponent for one-dimensional maps using a measure-preserving polynomial basis
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by Philip J. Aston and Oliver Junge PDF
Math. Comp. 83 (2014), 1869-1902 Request permission

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.
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Additional Information
  • Philip J. Aston
  • Affiliation: Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom
  • Email: P.Aston@surrey.ac.uk
  • Oliver Junge
  • Affiliation: Technische Universität München, Zentrum Mathematik, Boltzmannstr. 3, D-85747 Garching, Germany
  • Email: oj@tum.de
  • Received by editor(s): November 23, 2011
  • Received by editor(s) in revised form: September 17, 2012
  • Published electronically: November 12, 2013
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 83 (2014), 1869-1902
  • MSC (2010): Primary 37M25, 65P20
  • DOI: https://doi.org/10.1090/S0025-5718-2013-02811-6
  • MathSciNet review: 3194133