In this paper, the Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay was explored. First, the conditions of the occurrence of Hopf-zero bifurcation were obtained by analyzing the distribution of eigenvalues in correspondence to linearization. Second, the stability of Hopf-zero bifurcation periodic solutions was determined based on the discussion of the normal form of the system, and some numerical simulations were employed to illustrate the results of this study. Lastly, the normal form of the system on the center manifold was derived by using the center manifold theorem and normal form method.

中文翻译：

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环形单向耦合Toda振荡器的Hopf零分叉

本文研究了具有延迟的环形单向耦合Toda振荡器的Hopf零分支。首先，通过分析对应于线性化的特征值的分布来获得出现霍普夫零分叉的条件。其次，在讨论系统的正规形式的基础上，确定了Hopf-零分岔周期解的稳定性，并利用一些数值模拟来说明本研究的结果。最后，利用中心流形定理和法向形式方法推导了系统在中心流形上的范式。