Generation of Oscillatory Trajectories with Specified Stability Degree Using Recurrent Neural Networks

In the biological systems there are numerous examples of autonomously generated periodic activities. Several different periodic patterns are generated simultaneously in a living body. This paper discusses a method of generating periodic oscillatory trajectories in an artificial neural network. We propose a synthesis method of a neural network which generates desired autonomous periodic trajectories. The proposed method makes it possible to generate not only one periodic trajectory but also multiple different trajectories in a neural network simultaneously and to make each periodic trajectory possess specified stability degree. It is known that stability and stability degree of periodic trajectories (limit cycles) can be estimated by checking eigenvalues of Jacobian matrix of the Poincaré map defined on them. We propose a learning method of neural networks which makes the Poincaré map of each generated periodic trajectory possess specified stable eigenvalues. Experimental examples are presented to demonstrate the applicability and performance of the proposed method.