神经网络动力系统吸引子对称性的变化

用余维２的双Ｈｏｐｆ分叉的规范形方程设计了期望存储振荡型记忆模式的模拟四阶关系神经网络，所设计的网络向量场具有中心对称性。研究表明该网络发生二阶双Ｈｏｐｆ分叉后可以出现不变二环面。观察二环面上的销相运动，发现了系统出现对称破裂的规律，邓把双Ｈｏｐｆ分叉的两个频率比表示成既约分数的形式，当该分数的分子和分母均为奇数时，网络的吸引子保持对称性，而当分子主分母中任一个为偶数时，就会发生对称破裂。

SYMMETRICAL CHANGES OF ATTRACTORS IN THE DYNAMICAL SYSTEM OF THE NEURAL NETWORK

In this paper, an analog of the neural network of the fourth order aiming to store the oscillatory memory patterns is designed using the normal form equations for double Hopf bifurcation with two remaining dimensions. The vector field of the designed network is centrosymmetric. Results show that an invariant two-torus may be observed after the secondary Hopf bifurcation has happened. Investigating the phase-locking on the torus, we found the regularity of the symmetrical breaking. If the ratio of the freqencies of the double Hopf bifurcation is represented by an irreducible fraction, when both the numerator and the denominator are odd, the stable limit cycle possesses the full symmetry of the system. In the case that either the numerator or the denominator is even, symmetrical breaking will happen.