Forecasting of the chaos by iterations including multi-layer neural-network

We discuss a general method to forecast movements of the chaos. The method is based on functions of multi layer neural networks and a recurrent representation of functions. We are sure that forecasting of the chaos is an important problem. Movements of the chaos are extremely complex, and similar to natural phenomena. The forecasting for the chaos has been studied based on the embedding theory. They are called "one-step prediction". The embedding theory is effective and useful, and the results are accurate. But we wish to know long-term futures in practical objects. We need to take one step forward from the embedding theory. Our target is the long-term forecasting.