Solution of the inverse frobenius-perron problem for semi-Markov chaotic maps via recursive Markov state disaggregation

Abstract

Abstract-A novel solution of the inverse Frobenius-Perron problem for constructing semi-Markov chaotic maps with prescribed statistical properties is presented. The proposed solution uses recursive Markov state disaggregation to construct an ergodic map with a piecewise constant invariant density function that approximates an arbitrary probability distribution over a compact interval. The solution is novel in the sense that it provides greater freedom, as compared to existing analytic solutions, in specifying the autocorrelation function of the semiMarkov map during its construction. The proposed solution is demonstrated by constructing multiple chaotic maps with invariant densities that provide an increasingly accurate approximation of the asymmetric beta probability distribution over the unit interval. It is demonstrated that normalised autocorrelation functions with components having different rates of decay and which alternate in sign between consecutive delays may be specified. It is concluded that the flexibility of the proposed solution facilitates its application towards modelling of random signals in various contexts.

The Construction of Ergodic Maps with Prescribed Invariant Density and Power Spectrum Poles

“…The methods in [6], [7], [9] generalize the matrix solution of IFPP-I first proposed in [12]. Each method constructs a stochastic matrix with prescribed characteristics and which qualifies as the order-zero FPM of a PWL semi-Markov map.…”

“…The method proposed in [6] constructs a semi-Markov map with invariant density that approximates a prescribed invariant density and a PSD with an arbitrary number of poles, such that nonrepeating real values may be prescribed for a subset of the poles. It uses the observation that the unique nonzero and nonunity (in absolute value) eigenvalues of the map's order-zero FPM appear as poles in its PSD.…”

“…Furthermore, unlike [8], [9], the proposed method does not rely on a numerical search to find the map, and expressions are derived that qualify a candidate collection of poles as realizable. However, similar to recent methods that realize rational PSDs [6], [7], the proposed method does not afford the opportunity to select the zeros of the map's PSD. This implies the rational PSD of the map cannot be fully prescribed using these methods.…”

Solving the inverse Frobenius-Perron problem using stationary densities of dynamical systems with input perturbations

“…These include an entirely graph-theoretic method introduced in [13] to construct piecewise linear transformations that have piecewise constant stationary densities of relative minima value zero; a more generalized approach given in [14] to deriving an explicit relationship between stationary density functions and a special class of piecewise linear transformations over a partition of interest, and furthermore a matrix-based solution to construct 3-band transformations that preserves a given piecewise constant stationary density. Using this matrix-based algorithm, a recursive Markov state disaggregation scheme was proposed in [15] to construct multiple semi-Markov chaotic maps with stationary densities. The inverse problem was investigated in [4] for a class of symmetric maps that have stationary beta density functions and a unique solution can be obtained under the symmetry constraints of the unknown maps.…”

“…In [28] not only stationary density but also a temporal sequence of densities generated by the underlying system are used to estimate the unique transformation, and in [29,30] chaotic maps are determined from sequences of generated densities, which are both under the assumption that the evolution of an arbitrary density function is observable. This is, however, not always the case in practice, particularly as they require selecting initial conditions to generate densities [15].…”