We propose an analytical method for constructing ergodic and piecewise linear hat maps with a piecewise constant invariant density selected to approximate a prescribed invariant density, and a rational power spectral density with an arbitrary number of prescribed nonrepeating poles. These semi-Markov maps permit a matrix representation of the Frobenius-Perron operator, referred to as the Frobenius-Perron matrix. We prove that the poles of a piecewise linear hat map’s power spectral density coincide with the unique nonzero and nonunity (in absolute value) eigenvalues of the map’s Frobenius-Perron matrix. A  hat map is constructed such that it possesses a Frobenius-Perron matrix with a selectable eigenspectrum, thereby allowing for the prescription of its poles. A topological conjugate of this map is subsequently constructed. The conjugate map’s poles are equal to those of the original map, but its piecewise constant invariant density is selected to approximate the prescribed invariant density. Similar to previous methods, the novel method does not afford the opportunity to prescribe the zeros of the map’s power spectral density. However, unlike these methods, all of the poles are prescribable and no numerical search is performed during construction. We characterize several piecewise linear hat maps constructed using the novel method.

We present a direct solution to the problem of constructing a stochastic matrix with prescribed eigenspectrum, widely referred to as the stochastic inverse eigenvalue problem. The solution uses Markov state disaggregation to construct a Markov chain with stochastic transition matrix possessing the required eigenspectrum. Existing solutions that follow the same approach are limited to constructing matrices with real-valued eigenspectra only. The novel solution directly constructs matrices with complex-valued eigenspectra by applying a new disaggregation technique in tandem with a technique from a previous solution. Due to this generalization, the novel solution is able to successfully model physical systems from a larger family. Furthermore, the novel solution constructs the matrix in a finite and predetermined number of iterations, and without numerical approximation. The solution is demonstrated by deriving an expression for a set of 4 x 4 stochastic matrices sharing the same prescribed complex-valued eigenspectrum and indexed by a real parameter.

This paper presents a novel solution to the inverse Frobenius-Perron problem of reconstructing an unknown nonlinear and ergodic map from causal sequences of probability density functions generated by the map. The original solution to this problem successfully reconstructs members of the canonical map class (i.e., a subset of the piecewise linear semi-Markov maps), provided all the map’s branches are monotonically increasing. The original solution constructs a matrix estimate of the map’s Frobenius-Perron operator, which governs the evolution of density functions under iteration of the map, from the density sequences. The one-dimensional map is reconstructed from this matrix. In contrast, the proposed solution constructs a higher-order matrix estimate of the Frobenius-Perron operator. A member of the newly proposed class of generalized hat maps, a superset of the canonical maps, is constructed from this matrix estimate. The proposed solution successfully distinguishes between increasing and decreasing map branches and enlarges the class of maps that can be successfully reconstructed to canonical maps with any subset of decreasing branches. When used to reconstruct any piecewise linear semi-Markov map, the proposed solution generates a map with consistent invariant density and power spectrum mode characteristics, regardless of the unknown map’s canonicity or branch monotonicity. Numerical examples that illustrate the proposed solution’s characteristics are presented.