# American Institute of Mathematical Sciences

October  2021, 26(10): 5337-5354. doi: 10.3934/dcdsb.2020346

## Qualitative analysis of a generalized Nosé-Hoover oscillator

 1 School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou, Henan 450046, China 2 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China 3 Hubei Key Laboratory of Engineering Modeling and Science Computing, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

* Corresponding author: Xiao-Song Yang

Received  November 2019 Revised  July 2020 Published  October 2021 Early access  November 2020

Fund Project: The first author is supported by NSFC grant 51979116

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 $\mathbb R^3$.

Citation: Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5337-5354. doi: 10.3934/dcdsb.2020346
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##### References:
The grid is part of $S_{1}$ and the shadow is part of $S_{2}$
The shadow is the projection of the region $I$ on the plane $z = 0$
$A_{1}\rightarrow A_{2}$ means there are solutions from $A_{1}$ to $A_{2}$, $B_{1}\dashrightarrow A_{2}$ means there are solutions from $B_{1}$ to $A_{2}$ and these solutions have intersection with $X$-axis or $Y$-axis
From right to left are $l_{10}$ and $l_{20}$
From right to left are $l_{01}$, $l_{02}$, $l_{03}$ and $l_{04}$
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