Fractal Variation of Attractors in Complex Valued Neural Networks with Negative Resistive Nonlinearity

The complex valued neural networks are the extended version of conventional real valued neural networks Input and output signals weighting factors and neuron nonlinear functions are determined using complex number so that the information geometry of the network is constructed in complex space This feature is advantageously used especially for learning and expressing smooth dynamical attractors On the other hand chaotic aspects in biological and arti cial neural networks have become very attractive when we deal with time sequential behaviour of the networks Such non deterministic characteristics of the neural networks are sometimes related to spontaneous processes in brains or probabilistic operations of arti cial networks In many cases the chaotic behaviour originates from negative resistive nonlinearity In this paper a fractal variation of attractors in complex valued neural networks is reported When a negative resistive nonlinearity is introduced in a complex valued associative memory having circular attractors the attractor get unstabler in their phase relation As a looser nonlinearity becomes continuously a steeper one the attractor makes hops to various unstable states Between the parameter ranges of circular and non deterministic attractors it is found that the attractor bifurcation diagram shows a fractal variation This result suggests that the complex valued neural networks can be e ectively used as chaotic neural networks in coherent information processing such as coherent optical computing