[en] Evolution of probability densities generated by many practical dynamical systems can be more conveniently observed than individual point trajectories. This paper introduces a new method to reconstruct the unknown transformation of a 1D discrete-time dynamical system that is driven by an external control input with a given probability density function, using multiple sequences of converging probability densities generated by the perturbed underlying system. Regardless of different initial conditions the generated densities are demonstrated possessing strong convergence to a unique invariant density. Numerical simulation results validate the applicability of the developed algorithm as well as the performance in the presence of stochastic noise. (paper)