讨论了一类带有惯性项的时滞神经网络模型的Hopf分岔。首先从模型特征方程入手,分析了特征方程特征根的分布情况;结合已有文献中对系统平衡点稳定性的分析,得到了平衡点失稳后发生Hopf分岔的条件;利用伪振子分析法研究了平衡点在临界点附近的局部动力学行为,包括产生Hopf分岔的分岔方向及分岔周期解的稳定性,给出了分岔周期解的振幅估计的计算式;最后,通过计算机软件和数值模拟试验给出了平衡点在临界点附近的时间历程图或相图,很好地验证了前边对于稳定性分析,以及伪振子分析法对该模型在临界点附近产生的局部动力学行为研究的正确性。特别地,与原文献所采用的规范型方法相比较而言,伪振子分析法无论是在计算过程还是在计算结果以及计算结果的精确性上,都显示出其简便、快捷、准确和易于操作的特点。

The stability and the Hopf bifurcation of a delayed neural network model with an inertial term are investigated in this paper. The characteristic equation of the linearized time delay equation is first considered about the trivial solution, and the condition for the existence of a Hopf bifurcation is obtained based on the study of the root location of the characteristic equation. Then, the newly developed method, the pseudo-oscillator analysis, is applied to study the local dynamics round the trivial solution and near the bifurcation point, including the bifurcation direction, the stability of the bifurcation-induced periodic solution, and the estimation of the amplitude of the periodic solution. Finally, two case studies are given to validate the theoretical prediction of the bifurcated periodic solution, which is checked numerically by the plots of the time history, the phase portraits and the bifurcation diagrams. It is shown that the Hopf bifurcation is supercritical, and the bifurcation-induced periodic solution is stable. In addition, the results show that the pseudo-oscillator analysis has several advantages over the center manifold reduction and the normal form theory used in literature: it involves easy calculation about integration of harmonic functions only, it results in simple computational results in terms of the system parameters, and it offers an estimation of the bifurcated periodic solution with a high computational accuracy.