Hierarchy of Chaotic Maps with an Invariant Measure and their Compositions

Abstract

We give a hierarchy of many-parameter families of maps of the interval [0, 1] with an invariant measure and using the measure, we calculate Kolmogorov-Sinai entropy of these maps analytically. In contrary to the usual one-dimensional maps these maps do not possess period doubling or period-n-tupling cascade bifurcation to chaos, but they have single fixed point attractor at certain region of parameters space, where they bifurcate directly to chaos without having period-n-tupling scenario exactly at certain values of the parameters.

Hierarchy of random chaotic maps with an invariant measure

“…It should be mentioned that for trigonometric chaotic maps [8], their composition [9] and their coupling [7] the eigenstate of PF operator L corresponding to largest eigenvalues has already been obtained in our previous papers. Now,we choose the hierarchy of trigonometric chaotic maps Φ(N i , α, x), as the ensemble of chaotic maps.…”

“…Even thought one can define many-parameters random trigonometric chaotic maps with an invariant measure, but for simplicity we restrict ourselves here in this paper to twoparameters ones [9]. Random two-parameters trigonometric chaotic maps are defined as:…”

“…Additionally, random maps are of interest as models of convection by temporarily irregular fluid flows [13]. In this paper we consider random maps where, on each iteration, the map function Φ(x, α) is chosen randomly from a hierarchy of chaotic maps with an invariant measure which are introduced in our previous papers [8,9,10]. There are two noticeable advantages of random chaotic maps that are presented through this article.…”

“…Usually the analytic calculation of invariant measure of dynamical systems is a nontrivial task, hence there are limited number of maps with invariant measure like, Ulam-Von Neumann map [1], chebyshev maps [2], Katsura-Fukuda map [3], piecewise parabolic map [4], Tent map [5], Elliptic map [6] and finally hierarchy of one and manyparameter families of random trigonometric chaotic and one-parameter random elliptic chaotic maps of cn type and their coupling [8,9,7,10]. Here in this work we give a new hierarchy of random chaotic maps with an invariant measure, where using the invariant measure we discuss analytically the transition to chaos in these random dynamical systems.…”

“…Here in this paper we try to constructs hierarchy of random chaotic maps with an invariant measure by choosing the ensemble of one and many parameters of chaotic maps of trigonometric and elliptic types of references [8,9,10] as follows.…”