A Novel Approach to Solving the Generalized Inverse Frobenius-Perron Problem

Abstract

A new approach to solving a more general formulation of the inverse Frobenius-Perron problem, which requires the construction of a one-dimensional ergodic map with prescribed invariant probability density function and power spectral density, is presented. The proposed approach relies on a novel technique for generating distinct maps with the same invariant density, and which facilitates selection of the structural characteristics of each map in advance. We consider a new class of maps constructed with this technique, the piecewise monotonic hat maps, and present an algorithm for selecting the map parameters to achieve simultaneous and independent prescription of the invariant density and multimodal power spectrum characteristics. This approach to solving the generalized inverse Frobenius-Perron problem is demonstrated by constructing several ergodic maps with the beta invariant density as well as unimodal and bimodal power spectra with distinct mode center frequencies and bandwidths. We conclude that the proposed approach provides a means for generating more realistic models of systems and processes as compared to existing methods.