Qualitative analysis of a generalized Nosé-Hoover oscillator

In this paper, we analyze the qualitative dynamics of a generalized Nosé-Hoover oscillator with two parameters varying in certain scope. We show that if a solution of this oscillator will not tend to the invariant manifold \begin{document}$ \{(x,y,z)\in \mathbb R^3|x = 0,y = 0\} $\end{document}, it must pass through the plane \begin{document}$ z = 0 $\end{document} infinite times. Especially, every invariant set of this oscillator must have intersection with the plane \begin{document}$ z = 0 $\end{document}. In addition, we show that if a solution is quasiperiodic, it must pass through at least five quadrants of \begin{document}$ \mathbb R^3 $\end{document}.