规范形网络中的混沌吸引子

讨论多余维Hopf分叉三阶规范形的普适开折形成的网络更进一步的复杂动力学行为．通过对余维二Hopf分叉的规范形网络多级分叉的分析，发现在参数空间的某个区域会出现二环面，将S形非线性加入规范形网络，在出现二环面的区域内可以出现混沌．本文给出了该混沌吸引子的相图及其二阶Poincare映射的图景．由这些图可以看到该混沌吸引子具有非常奇妙的形态：某些二阶Poincare映射像一只逼真的蝴蝶．

A CHAOTIC ATTRACTOR IN THE NORMAL FORM NETWORK

Biological experiments of mammalian brain have shown that real neural systems exhibit a range of phenomena such as oscillations, phase-locking and even chaos. The chaotic behaviors simulate the information processing mechanisms of the real neural systems at a higher level. In this paper the bifurcation and chaos of the high order correlation networks will be studied.In some previous discussions about the high order correlation neural networks, we learned that the high order correlation networks expected to store static and oscillating memory patterns could be designed and synthesized by the normal form for the pitchfork and Hopf bifurcation. The networkformed by the normal form equations is called the normal form network. The transformationused in the proving process of designing the high order correlation networks with the normal form networks is inversible linear transformation. Thus the vector field of the high order networks istopologically equivalent to that of the normal form networks. We can study the complex dynamicalbehaviors of the normal form networks in order to understand that of the high order networks. Inthis paper the complex dynamical behaviors of the normal form network formed by the universalunfoldings of the normal form for the double Hopf bifurcation with codimension 2 are discussed.After analyzing the secondary Hopf bifurcations of the network, we observed the motions on twotorus in some areas of the parameter space. If the sigmoid nonlinearity is added to the right hand side of the equations of the network, chaos may be presented in the same parameter regime. The phase portrait and the second order Poincare maps of the attractor are given. The chaotic attractor displays a curious structure on some second order Poincare maps. It looks like a butterfly. This indicates that chaos with fractal structure can mimic nature.l) The project supported by the National Natural Science Foundation of China (No.19702015).