We consider the four-dimensional hyperchaotic system $\dot{x}=a(y-x)$, $\dot{y}=bx+u-y-xz$, $\dot{z}=xy-cz$, and $\dot{u}=-du-jx+exz$, where a, b, c, d, j, and e are real parameters. This system extends the famous Lorenz system to four dimensions and was introduced in Zhou et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 27, 1750021 (2017). We characterize the values of the parameters for which their equilibrium points are zero-Hopf points. Using the averaging theory, we obtain sufficient conditions for the existence of periodic orbits bifurcating from these zero-Hopf equilibria and give some examples to illustrate the conclusions. Moreover, the stability conditions of these periodic orbits are given using the Routh–Hurwitz criterion.

中文翻译：

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四维超混沌系统的零霍普夫分岔

我们考虑四维超混沌系统 $\dot{X}=\mathrm{一种}(是-X)$, $\dot{是}=乙X+你-是-Xz$, $\dot{z}=X是-Cz$， 和 $\dot{你}=-d你-jX+\mathrm{电子}Xz$，其中a，b，c，d，j和e是实参。该系统将著名的洛伦兹系统扩展到四个维度，并在 Zhou等人中引入。，国际。J.分岔混沌应用。科学。英。27 , 1750021 (2017)。我们描述了其平衡点是零-Hopf 点的参数值。利用平均理论，我们从这些零-Hopf 平衡中得到了周期性轨道存在分叉的充分条件，并给出了一些例子来说明结论。此外，这些周期轨道的稳定性条件是使用 Routh-Hurwitz 准则给出的。