In this paper, the profile and stability of alternating wave solution, which arises as a bifurcated periodic solution of equivariant Hopf bifurcation with amazing properties, are investigated for a Stuart-Landau system consisting of three oscillators. The method of multiple scales is used to compute the normal form equation up to fifth order. The Floquet theory is introduced because it is difficult to directly analyze the stability of the alternating wave solution. By applying a time-varying complex coordinate transformation which does not change the stability of the solution of normal form that represents the alternating wave, the multipliers that completely determine the stability of alternating wave solution are explicitly solved. As a result, the criteria on parameters such that stable alternating wave solutions can be observed are provided. Based on studies through examples, we show that the proposed scheme of analysis is effective and some results on how parameters influence the stability of the alternating wave solution can be summarized. Our analysis confirms Golubitsky's assertion that the alternating wave solution will not be stable immediately after the equivariant Hopf bifurcation. We also find that a large time delay and a complex nonlinear gain will enhance the stability of alternating wave solution.

在本文中，研究了由三个振荡器组成的Stuart-Landau系统的交替波解的轮廓和稳定性，它是具有等价性质的Hopf分叉的分叉周期解的分支周期解。多尺度方法用于计算法线形式方程，直到五阶。引入浮球理论是因为难以直接分析交变波解的稳定性。通过应用时变复坐标变换，该变换不会改变表示交变波的法线形式解的稳定性，因此可以明确解决完全确定交变波解的稳定性的乘数。结果，提供了关于参数的标准，使得可以观察到稳定的交变波解。通过实例研究，表明所提出的分析方案是有效的，并且可以总结出有关参数如何影响交变波解的稳定性的一些结果。我们的分析证实了Golubitsky的断言：等变的Hopf分叉之后，交变波解将不稳定。我们还发现，较大的时延和复杂的非线性增益将增强交变波解的稳定性。