Probability properties of one-dimensional piece-wise linear chaotic map having two linear brunches (Rényi map) are
investigated. The map dynamics depends on a parameter defining substantially view of the map, i.e. slopes of map
linear branches and proportion between them. This dependence is very sensitive, and there is the infinite set of parameter
values providing existence of piecewise constant invariant density of the map. These values of the parameter may be
obtained by solving corresponding algebraic equation. These properties allow us to apply the map for modeling complex
chaotic regimes by means of switching between various values of parameter. The map is suggested to be suitable for
description of degrees of chaotic neuron reactions on weak excitations and for chaotic encryption.
In the paper, the analytical method of constructing special generating functions for eigenfunctions and eigenvalues of the Perron-Frobenius operator corresponding to piece-wise symmetric one-dimensional chaotic maps is justified. Some properties of eigenfunctions are illustrated. An extension of the results for maps related with piece-wise ones by invertible nonlinear transformations is showed. The results for chaotic one-dimensional maps modeling biological and physiological rythmes (neuron activity or heart beats) and having invariant distributions in the form of various types of exponential law (standard distribution and its generalizations) are presented.
In the paper, some chaotic one-dimensional maps modeling biological and physiological rythmes (neuron activity or heart beats) are suggested. These maps are constructed as transformations that are topologically conjugated to piecewise linear ergodic and mixing endomorphism having the uniform invariant distributions and exact trajectory characteristics (as functions of initial conditions and number of iterations). New conjugated maps are defined on infinite intervals. They have invariant distributions in the form of various types of exponential law (standard distribution and its generalizations). It is shown that dynamics of chaotic generator depend on the choice of the basic endomorphism. Hence, there are the countable set of generators of chaotic rythmes that have the same invariant distribution and conjugation function, but obtain various rates of convergence to the invariant distribution and various autocorrelation functions. Expressions for named characteristics of the chaotic generators are derived.
A model of a scattering biological 3D medium with irregularities in the form of the monoorientated cylindrical particles is considered. It is assumed that these scatterers have random geometrical parameters and randomly located in the space with Poisson statistics. Their optical parameters differ from the corresponding background values. The characteristic functional of such random field and its derivatives that give the expressions for the mean value and autocorrelation function are analytically obtained. The Wiener- Khinchine's spatial spectrum is calculated too. The obtained expressions contain the statistical moments of geometrical and optical parameters of scattering centers and their space density.
In the work the medium inhomogeneities are considered as the optical parameter deviations from a certain mean values and represented by a random spatial 'pulse' process. These 'pulses' are supposed to have arbitrary geometrical forms, random parameters and random locations and orientations in space. We obtain the general analytical representation for the characteristic functional, autocorrelation function and Wiener-Khinchin's spectrum of the modeling process, that are applicable to various geometry of scattering objects and may be easy calculated. The corresponding relations contain statistical moments of geometrical and optical parameters of scattering centers and their spatial density. As an example the obtained relations are written for the medium with the spheroidal irregularities. The introduced model of the random continuous scattering medium may be useful in the classification of the solutions of the inverse problem of light interactions with homogeneous medium and in the noninvasive diagnostics.