IDENTIFICATION OF CHAOTIC DYNAMICAL-SYSTEMS WITH BACKPROPAGATION NEURAL NETWORKS

In this paper, we clarify fundamental properties of conventional back-propagation neural networks to learn chaotic dynamical systems by some numerical experiments. We train three-layers networks using back-propagation algorithm with the data from two examples of two-dimensional discrete dynamical systems. We qualitatively evaluate the trained networks with two methods analysing geometrical mapping structure and reconstruction of an attractor by the recurrent feedback of the networks. We also quantitatively evaluate the trained networks with calculation of the Lyapunov exponents that represent the dynamics of the recurrent networks is chaotic or periodic. In many cases, the trained networks show high ability of extracting mapping structures of original two-dimensional dynamical systems. We confirm that the Lyapunov exponents of the trained networks correspond to whether the reconstructed attractors by the recurrent networks are chaotic or periodic.