The Zero–Hopf bifurcations of the quadratic polynomial differential jerk systems in \({\mathbb {R}^3}\)

Abstract

We study the zero–Hopf bifurcations of all quadratic polynomial differential jerk systems in \({\mathbb {R}^3}\)

$$\begin{aligned} \begin{array}{l} \dot{x}=y,\\ \dot{y}=z,\\ \dot{z}=a_{0}+a_{1}x+a_{2}y+a_{3}z+a_{4}x^{2}+a_{5}xy+a_{6}xz+a_{7}y^{2}+a_{8}yz+a_{9}z^{2}, \end{array} \end{aligned}$$

where the dot denotes derivative with respect to the independent variable t and the coefficients \(a_{k}\), for \(k=0,1,...,9\), are real.