Reconstruction of one-dimensional chaotic maps from sequences of probability density functions

In many practical situations, it is impossible to measure the individual trajectories generated by an unknown chaotic system, but we can observe the evolution of probability density functions generated by such a system. The paper proposes for the first time a matrix-based approach to solve the generalized inverse Frobenius–Perron problem, that is, to reconstruct an unknown one-dimensional chaotic transformation, based on a temporal sequence of probability density functions generated by the transformation. Numerical examples are used to demonstrate the applicability of the proposed approach and evaluate its robustness with respect to constantly applied stochastic perturbations.