四神经元时滞网络的稳定性与分叉
茅晓晨，胡海岩
摘要:本文研究由四个神经元构成的时滞网络的动力学。首先分析网络平衡点的数目,给出网络发生静态分叉的条件。以时滞为分叉参数,通过研究网络平衡点处线性化系统的特征方程,得到网络平衡点全时滞稳定的参数区域和与时滞相关的稳定区域,并给出网络平衡点失去稳定性发生Hopf分叉的条件。采用中心流形定理和Normal Form规范型理论对Hopf分叉周期解的性质进行分析,得到判断分叉方向、分叉解稳定性和分叉解周期的公式,最后进行数值仿真验证理论分析的结果。
关键词：神经网络；时滞；分叉；周期运动
Stability and Bifurcation Analysis of a Four-neuron Network with Delays
Xiaochen Mao, Haiyan Hu
Abstract：The stability and bifurcation of a delayed network of four neurons with a short-cut connection were revealed. The system can be represented by a set of nonlinear differential equations with multiple delays. The sufficient conditions for delay-independent and delay-dependent asymptotic stability of trivial equilibrium in local sense were addressed by discussing the distribution of characteristic roots of linearized system. By analyzing the equilibrium point number and stability of the network, the existence of a steady state bifurcation was shown. The sufficient conditions for the periodic responses arising from a Hopf-bifurcation with respect to the delay at the trivial equilibrium were obtained. The features of the bifurcated periodic responses depend on the nonlinearity of the system. The explicit formulae determining the direction, stability and period of the bifurcated periodic responses were given on the basis of center manifold theory and normal form approach. Numerical simulations were given to validate the theoretical analysis. It shows that the short-cut connection plays an important role in the dynamics of network.
Keyword：neural network; delay; bifurcation; periodic response
原文链接：http://d.wanfangdata.com.cn/periodical/lxjk200901001
