The Hopf bifurcation analysis on a time-delayed recurrent neural network in the frequency domain

2000 MSC: 92B20 34C23 37C27 93C80 Keywords: Recurrent neural networks Distributed delays Hopf bifurcation Frequency domain Generalized Nyquist stability criterion a b s t r a c t In this paper, a class of recurrent neural networks with distributed delays and a strong kernel is studied. It is shown that the Hopf bifurcation occurs as the bifurcation parameter, the mean delay, passes a critical value where a family of periodic solutions emanates from the equilibrium. The existence and stability of such solutions are determined by the Hopf bifurcation theorem in the frequency domain and the generalized Nyquist stability criterion. 1. Introduction and problem statement Recurrent neural networks (RNNs) including Hopfield neural networks and cellular neural networks have been used extensively in different areas such as signal processing, pattern recognition, optimization and associative memories. They are also one of the best choices for learning input and output data which dynamically vary by time. One of the main focuses of research on RNNs has always been the existence of periodic solutions and mechanisms under which such solutions emerge. One of the well-known reasons behind the emergence of periodic solutions in RNNs is the presence of time delays. In this paper, we study a three-node recurrent neural network with distributed delays. Ruiz et al. studied this model for the first time without time delays in [18] where a particular configuration of a recurrent neural network, illustrated in Fig. 1, was considered. In Fig. 1, uðtÞ is the input and yðtÞ is the output of the network. This recurrent neural network can be described by the following system: