Construction of semi-Markov ergodic maps with selectable spectral characteristics via the solution of the inverse eigenvalue problem

Abstract

This paper presents a novel technique for constructing semi–Markov ergodic maps that harnesses a new solution of the inverse eigenvalue problem for 3–by–3 doubly stochastic matrices. The proposed solution facilitates the selection of the spectral characteristics of the trajectories generated by these ergodic maps through the selection of appropriate eigenvalues for the Frobenius–Perron matrices associated with the maps. It is proved that the proposed solution is able to realise all possible eigenvalue triples for the matrices of interest, thereby providing greater freedom in selecting the power spectral density than existing techniques. The novel technique is demonstrated by constructing several semi–Markov ergodic maps with distinct power spectra. It is concluded that the flexibility and versatility of the technique holds potential for the purpose of system modelling in various contexts.