Solving the inverse Frobenius-Perron problem using stationary densities of dynamical systems with input perturbations

•One-dimensional discrete-time dynamical systems are inferred from stationary densities generated by the systems in the presence of input perturbation.•The main assumption is that the stationary densities can be observed and estimated given arbitrary initial conditions.•The unique stationary density function generated by a perturbed system exists, corresponding to a specified input density function.•The stationary density is dependent on the probability density function of an external control input given a transformation.•A practical algorithm of determining the unknown transformation using stationary densities generated by a system with linearly independent input density functions is introduced.