Symmetry-Breaking and Chaos in Oscillatory Neural Networks
Yonghong Chen, Jian-Xue Xu, Fang Tong · 2001
Abstract Complex dynamical behavior of neural networks may lead to new methodology of information processing. In this paper the dynamics of a neural network designed by the normal form for Hopf bifurcation is studied. The secondary Hopf bifurcation of the network is discussed and a two-torus is observed. Examining the phase-locking motions on the two-torus, we present the conditions of symmetry-breaking occurring in the system. If the ratio of the two frequencies of the codimension two Hopf bifurcation is represented by an irreducible fraction, then the symmetry-breaking will occur when either the numerator or the denominator of the fraction is an even number. Chaotic attractors may be created with the sigmoid nonlinearities added to the right hand side of the normal form equations. The phase trajectory and the second order Poincaré maps of the chaotic attractor are given. The chaotic attractor looks like a butterfly on some of the second order Poincaré maps. This is a marvelous example for chaos to mimic nature.