Creation and control of Fractal spatiotemporal hierarchical structure

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views the system as being closed, I.e. an autonomous system. Another views the system as being open, I.e. a non-autonomous system. The former approach assumes the external environment to be composed of constituent subsystems. In relation to this approach, many researchers have been interested in chaotic phenomena. Consequently, for the autonomous system, chaos has been established as a prominent research area that might be key to developing an understanding of complex systems in nature. On the other hand, the investigation into non-autonomous systems is still in its initial phase, and no clear theoretical framework has yet been established. We believe that the main reason that little research has thus far been done in this area is the difficulty in examining the interaction between systems. Therefore, we must attempt to determine how we can describe this interaction from a dynamical systems point of view. A theory for continuous dynamical systems stochastically excited by temporal exte … More rnal inputs has been presented. The theory suggests that the dynamics of continuous-time recurrent neural networks (RNNs) is generally characterized by a set of continuous trajectories with a fractal-like structure in hyper-cylindrical phase space. We refer to this dynamics as the fractal transition. Three types of numerical experiments are discussed in order to investigate the learning process and noise effects in terms of the fractal transition. First, to analyze how an RNN learns desired input {output transformations, a simple example with a single state was examined in detail. A fractal structure similar to a Cantor set was clearly observed in the learning process. This finding sheds light on the learning of RNNs, I.e. it suggests that the learning is a process of adjusting the fractal dimension. Second, input noise effects on the fractal structure were investigated. The results show that small-scale hierarchical structures are broken by noise. Third, using a network with twenty states, we show that fractal transition is a universal characteristic of RNNs driven by switching inputs. Less