Neural Networks and Chaos: Construction, Evaluation of Chaotic Networks, and Prediction of Chaos with Multilayer Feedforward Networks

Many research works deal with chaotic neural networks for various fields of application. Unfortunately, up to now these networks are usually claimed to be chaotic without any mathematical proof. The purpose of this paper is to establish, based on a rigorous theoretical framework, an equivalence between chaotic iterations according to Devaney and a particular class of neural networks. On the one hand we show how to build such a network, on the other hand we provide a method to check if a neural network is a chaotic one. Finally, the ability of classical feedforward multilayer perceptrons to learn sets of data obtained from a dynamical system is regarded. Various Boolean functions are iterated on finite states. Iterations of some of them are proven to be chaotic as it is defined by Devaney. In that context, important differences occur in the training process, establishing with various neural networks that chaotic behaviors are far more difficult to learn.