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Properties of distributions and correlation integrals for generalised versions of the logistic map

Hall, Peter A. & Wolff, Rodney C. (1998) Properties of distributions and correlation integrals for generalised versions of the logistic map. Stochastic Processes and Their Applications, 77(1), pp. 123-137.

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Abstract

We study a generalised version of the logistic map of the unit interval $(0,1)$, in which the point $x$ is taken to $1−|2x−1|^ u$. Here, $ u >0$ is a parameter of the map, which has received attention only when $ u =1$ and 2. We obtain the invariant density when $ u = frac12$, and derive properties of invariant distributions in all other cases. These are obtained by a mixture of analytic and numerical argument. In particular, we develop a technique for combining "parametric" information, available from the functional form of the map, with "non-parametric" information, from a Monte Carlo study. Properties of the correlation integral under the invariant distribution are also derived. It is shown that classical behaviour of this test statistic, which demands that the logarithm of the integral have slope equal to the lag, is valid if and only if $ u leq 2$.

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3 citations in Web of Science®

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ID Code: 5943
Item Type: Journal Article
Keywords: Chaos, correlation integral, invariant distribution, logistic map
DOI: 10.1016/S0304-4149(98)00010-6
ISSN: 0304-4149
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > PURE MATHEMATICS (010100)
Australian and New Zealand Standard Research Classification > PHYSICAL SCIENCES (020000) > CLASSICAL PHYSICS (020300)
Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > STATISTICS (010400)
Divisions: Current > QUT Faculties and Divisions > QUT Business School
Copyright Owner: Copyright 1998 Elsevier
Copyright Statement: Reproduced in accordance with the copyright policy of the publisher.
Deposited On: 05 Jan 2007
Last Modified: 05 Jan 2011 23:29

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