Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-22T09:31:02.554Z Has data issue: false hasContentIssue false
  • English
  • Français

A Maximum Entropy Method Based on Orthogonal Polynomials for Frobenius-Perron Operators

Published online by Cambridge University Press:  03 June 2015

Jiu Ding  and
Noah H. Rhee
Jiu Ding*
Affiliation:
Department of Mathematics, The University of Southern Mississippi, Hattiesburg, MS 39406-5045, USA
Noah H. Rhee*
Affiliation:
Department of Mathematics and Statistics, University of Missouri-Kansas City, Kansas City, MO 64110-2499, USA
*
Corresponding author. URL: http://r.web.umkc.edu/rheen/ Email: Jiu.Ding@usm.edu
Email: rheen@umkc.edu
Get access

Abstract

Let S: [0, 1]→[0, 1] be a chaotic map and let f* be a stationary density of the Frobenius-Perron operator PS: L1→L1 associated with S. We develop a numerical algorithm for approximating f*, using the maximum entropy approach to an under-determined moment problem and the Chebyshev polynomials for the stability consideration. Numerical experiments show considerable improvements to both the original maximum entropy method and the discrete maximum entropy method.

Keywords

41A3565D0765J10Frobenius-Perron operatorstationary densitymaximum entropyorthogonal polynomialsChebyshev polynomials
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 3 , Issue 2 , April 2011 , pp. 204 - 218
Copyright
Copyright © Global-Science Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Beck, C. and Schloögl, , Thermodynamics of Chaotic Systems, An Introduction, Cambridge University Press, 1993.Google Scholar
[2] Biswas, P., Shimoyama, H. and Mead, L., Lyaponov exponent and natural invariant density determination of chaostic maps: an iterative maximum entropy ansatz, preprint.Google Scholar
[3] Borwein, J. M. and Lewis, A. S., On the convergence of moment problems, Trans. Amer. Math. Soc., 325(1) (1991), pp. 249–271.Google Scholar
[4] Ding, J., A maximum entropy method for solving Frobenius-Perron operator equations, Appl. Math. Compt., 93 (1998), pp. 155–168.Google Scholar
[5] Ding, J. and Mead, L., Maximum entropy approximation for Lyaponov exponents of chaostic maps, J. Math. Phys., 43(5) (2002), pp. 2518–2522.Google Scholar
[6] Lasota, A. and Mackey, M., Chaos, Fractals, and Noises, Second Edition, Springer-Verlag, New York, 1994.Google Scholar
[7] Li, T. Y., Finite approximation for the Frobenius-Perron operator, a solution to Ulam’s conjecture, J. Approx. Theor., 17 (1976), pp. 177–186.Google Scholar
[8] Mead, L. R. and Papanicolaou, N., Maximum entropy in the problem of moments, J. Math. Phys., 25 (1984), pp. 2404–2417.Google Scholar
[9] Natanson, I. P., Constructive Function Theory, Frederick Ungar, New York, 1965.Google Scholar
[10] Ortega, J. M. and Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, 1970.Google Scholar
[11] Ulam, S., A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, No. 8, Interscience, New York, 1960.Google Scholar