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 $ \{(x,y,z)\in \mathbb R^3|x = 0,y = 0\} $, it must pass through the plane $ z = 0 $ infinite times. Especially, every invariant set of this oscillator must have intersection with the plane $ z = 0 $. In addition, we show that if a solution is quasiperiodic, it must pass through at least five quadrants of $ \mathbb R^3 $.

中文翻译：

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广义 Nosé-Hoover 振荡器的定性分析

在本文中，我们分析了具有在一定范围内变化的两个参数的广义 Nosé-Hoover 振荡器的定性动力学。我们证明如果这个振荡器的解不会趋向于不变流形 $ \{(x,y,z)\in \mathbb R^3|x = 0,y = 0\} $，它必须通过平面 $ z = 0 $ 无限次。特别是，该振荡器的每个不变量集必须与平面 $ z = 0 $ 相交。此外，我们证明如果一个解是准周期的，它必须至少通过 $\mathbb R^3 $ 的五个象限。