Let the three-dimensional differential system defined by the jerk equation $\stackrel{⃛}{x}=-a\ddot{x}+x{\dot{x}}^2-x^3-bx+c\dot{x}$, with $a,b,c\in \mathbb{R}$. When $a=b=0$ and $c<0$ the equilibrium point localized at the origin of coordinates is a zero-Hopf equilibrium. We analyse the zero-Hopf bifurcation occurring at this singular point after persuading a quadratic perturbation of the coefficients. Particularly, by using averaging theory of second order, we prove that up to three periodic orbits born as the parameter of the perturbation tends to zero.

中文翻译：

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3D冲击系统中的零霍夫分叉

令由jerk方程定义的三维微分系统 $\stackrel{⃛}{X}=-\mathrm{一种}\ddot{X}+X{\dot{X}}^2-X^3-bX+C\dot{X}$，带有 $\mathrm{一种}，b，C\in \mathbb{[R}$。什么时候$\mathrm{一种}=b=0$ 和 $C<0$位于坐标原点的平衡点是零霍夫平衡。在说服系数的二次扰动之后，我们分析了在此奇异点处发生的零霍夫分支。特别地，通过使用二阶平均理论，我们证明作为扰动参数而生的多达三个周期性轨道趋于零。