The Explicit Solutions of Frobenius-Perron Equation for the Chaotic Infinite Maps

Abstract

The simple one-dimensional mappings which demonstrate the chaotic behavior on the infinite interval and the appertaining explicit invariant densities having the forms of Cauchy distribution, F-distribution, and Z-distribution are presented.

Generating Different Statistical Distributions by the Chaotic Skew Tent Map

“…When a = 1/2, this skew tent map becomes the regular tent map studied in [Golubentsev & Anikin, 1998]. …”

“…Golubentsev and Anikin [1998] have shown that the tent map is useful in generating several familiar statistical distributions. In this paper, we extend the results of [Golubentsev & Anikin, 1998] to the skew tent map, which contains the tent map as a special case, and show further that virtually any statistical distribution can be generated via the skew tent map.…”

“…According to both [Golubentsev & Anikin, 1998;Hasler & Maistrenko, 1997], the time series generated from the following map:…”

The Schröder functional equation and its relation to the invariant measures of chaotic maps

“…where j runs over all the inverse branches of T . As we can see, this is the Frobenius-Perron equation for |q ′ (x)|, therefore, if there exist a function α(x) = |q ′ (x)| as a global solution of (14), defined on I, for |λ| = r, then the corresponding invariant measure of T is…”

“…In the last three decades a series of papers, devoted to deterministic chaos, are reporting examples of chaotic maps that are handled by means of a change of variables method [6,14,15,16,19,20,22,23,25,27,28,29]. Their study concerns with nonlinear piecewise transformations that are topologically conjugated, or semi-conjugated, to linear piecewise maps.…”