Riddled basins of the optimal states in learning dynamical systems

Abstract A new phenomenon called “riddled basin” has been drawing attention of nonlinear scientists. Riddled basin is an extremely complicated domain of attraction usually accompanying invariant sets for dynamical systems driven by chaotic signal, e.g. chaotic synchronizing systems. In this paper, the optimal state of a back-propagation learning of chaotic time series is shown to have a “locally riddled basin” by a numerical experiment. It is also rigorously proved that the optimal state of learning of the tent map by a gradient-descent method can be an attractor with a riddled basin.