Observing symmetry-breaking and chaos in the normal form network

The complex dynamical behaviors of neural networks may deduce new information processing methodology. In this paper, the dynamics of a normal form network with Z₂ symmetry is studied. The secondary Hopf bifurcation of the network is discussed and a two-torus is observed. Examining the phase-locking motions of the two-torus, we present the regularity 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, symmetry-breaking occurs when either the numerator or the denominator of the fraction is even. Chaotic attractors may be created with sigmoid nonlinearities added to the right-hand side of the normal form equations. The trajectory and 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 mimicking nature.