The solution of the inverse problem for the Perron-Frobenius equation

A description of stochastic systems using chaotic maps

“…We now consider the inverse Perron-Frobenius problem: suppose we are given a probability density function g(x) on R, can we find a transformation τ such that g(x) is the unique probability density function invariant under τ? This problem has been dealt with by Ershov and Malinetskiȋ [2] and in [3] from a computational perspective.…”

“…In this paper, we establish a method for describing flows of probability density functions by means of discrete-time chaotic maps. We start with a standard map whose invariant probability density function is known and then use it to derive other invariant probability density functions by a simple conjugation process which solves the inverse Perron-Frobenius problem [2,3] in a time-varying setting.…”

Constructing complete chaotic maps with reciprocal structures

“…We start with a general description of complete chaotic maps [2,8] defined on the unitinterval: I = [0,1]. The particular choice of the unit-interval as a map's domain is not restrictive since any chaotic map defined on a closed interval can always be transformed to a map defined on the unit-interval through variable substitution (linear topological conjugation).…”

On complete chaotic maps with tent-map-like structures

“…Recent advances along this endeavor include the differential equation approach [11], the transverse to conjugation approach [7,3], the branching function approach discussed in [4,8], and the stochastic approach adopted in [10]. While these studies greatly enhance our understanding of the ''micro'' relationship between the individual functional form and its related statistical features, the current research, however, aims to advance one-step further along the direction of exploring some genuine ''macro'' characteristics exhibited in some classes of chaotic maps that share identical analytical characteristics or similar statistical properties.…”