Ge Juhong, Xu Jian
School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China.
IEEE Trans Neural Netw. 2010 Mar;21(3):439-50. doi: 10.1109/TNN.2009.2038911. Epub 2010 Jan 29.
A bidirectional associative memory (BAM) neural network with four neurons and two discrete delays is considered to represent an analytical method, namely, perturbation-incremental scheme (PIS). The expressions for the periodic solutions derived from Hopf bifurcation are given by using the PIS. The result shows that the PIS has higher accuracy than the center manifold reduction (CMR) with normal form for the values of time delay not far away from the Hopf bifurcation point. In terms of the PIS, the necessary and sufficient conditions of synchronized periodic solution arising from a Hopf bifurcation are obtained and the synchronized periodic solution is expressed in an analytical form. It can be seen that theoretical analysis is in good agreement with numerical simulation. It implies that the provided method is valid and the obtained result is correct. To the best of our knowledge, the paper is the first one to introduce the PIS to study the periodic solution derived from Hopf bifurcation for a 4-D delayed system quantitatively.
一个具有四个神经元和两个离散延迟的双向联想记忆（BAM）神经网络被视为一种分析方法，即摄动增量法（PIS）。利用PIS给出了从Hopf分岔导出的周期解的表达式。结果表明，对于离Hopf分岔点不远的时间延迟值，PIS比具有范式的中心流形约化（CMR）具有更高的精度。基于PIS，得到了由Hopf分岔产生的同步周期解的充要条件，并以解析形式表示了同步周期解。可以看出，理论分析与数值模拟结果吻合良好。这意味着所提供的方法是有效的，所得到的结果是正确的。据我们所知，本文是第一篇引入PIS定量研究四维延迟系统中由Hopf分岔导出的周期解的文章。
