# 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 and Continuous Dynamical Systems - B, 2021, 26 (10) : 5337-5354. doi: 10.3934/dcdsb.2020346
##### References:
 [1] Q. Han and X.-S. Yang, Qualitative analysis of the Nosé-Hoover oscillator, Qual. Theory Dyn. Syst., 19 (2020), 1-36.  doi: 10.1007/s12346-020-00340-1. [2] W. G. Hoover, Canonical dynamics: Equilibrium phase-space distributions, Phys. Rev. A, 31 (1985), 1695-1697.  doi: 10.1103/PhysRevA.31.1695. [3] S. Nosé, A unified formulation of the constant temperature molecular dynamics methods, Journal of Chemical Physics, 81 (1984), 511-519. [4] S. Nosé, A molecular dynamics method for simulations in the canonical ensemble, Molecular Physics, 52 (2002), 255-268. [5] H. A. Posch, W. G. Hoover and F. J. Vesely, Canonical dynamics of the nosé oscillator: Stability, order, and chaos, Phys. Rev. A, 33 (1986), 4253-4265.  doi: 10.1103/PhysRevA.33.4253. [6] P. C. Rech, Quasiperiodicity and chaos in a generalized Nosé-Hoover oscillator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650170, 7 pp. doi: 10.1142/S0218127416501704. [7] J. C. Sprott, W. G. Hoover and C. G. Hoover, Heat conduction, and the lack thereof, in time-reversible dynamical systems: Generalized nosé-hoover oscillators with a temperature gradient, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 89 (2014), 042914-042914.  doi: 10.1103/PhysRevE.89.042914. [8] L. Wang and X.-S. Yang, The invariant tori of knot type and the interlinked invariant tori in the nosé-hoover oscillator, European Physical Journal B, 88 (2015), 1-5.  doi: 10.1140/epjb/e2015-60062-1. [9] L. Wang and X.-S. Yang, A vast amount of various invariant tori in the Nosé-Hoover oscillator, Chaos, 25 (2015), 123110, 6 pp. doi: 10.1063/1.4937167. [10] L. Wang and X.-S. Yang, The coexistence of invariant tori and topological horseshoe in a generalized Nosé-Hoover oscillator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750111, 12 pp. doi: 10.1142/S0218127417501115. [11] L. Wang and X.-S. Yang, Global analysis of a generalized Nosé-Hoover oscillator, J. Math. Anal. Appl., 464 (2018), 370-379.  doi: 10.1016/j.jmaa.2018.04.013.

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##### References:
 [1] Q. Han and X.-S. Yang, Qualitative analysis of the Nosé-Hoover oscillator, Qual. Theory Dyn. Syst., 19 (2020), 1-36.  doi: 10.1007/s12346-020-00340-1. [2] W. G. Hoover, Canonical dynamics: Equilibrium phase-space distributions, Phys. Rev. A, 31 (1985), 1695-1697.  doi: 10.1103/PhysRevA.31.1695. [3] S. Nosé, A unified formulation of the constant temperature molecular dynamics methods, Journal of Chemical Physics, 81 (1984), 511-519. [4] S. Nosé, A molecular dynamics method for simulations in the canonical ensemble, Molecular Physics, 52 (2002), 255-268. [5] H. A. Posch, W. G. Hoover and F. J. Vesely, Canonical dynamics of the nosé oscillator: Stability, order, and chaos, Phys. Rev. A, 33 (1986), 4253-4265.  doi: 10.1103/PhysRevA.33.4253. [6] P. C. Rech, Quasiperiodicity and chaos in a generalized Nosé-Hoover oscillator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650170, 7 pp. doi: 10.1142/S0218127416501704. [7] J. C. Sprott, W. G. Hoover and C. G. Hoover, Heat conduction, and the lack thereof, in time-reversible dynamical systems: Generalized nosé-hoover oscillators with a temperature gradient, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 89 (2014), 042914-042914.  doi: 10.1103/PhysRevE.89.042914. [8] L. Wang and X.-S. Yang, The invariant tori of knot type and the interlinked invariant tori in the nosé-hoover oscillator, European Physical Journal B, 88 (2015), 1-5.  doi: 10.1140/epjb/e2015-60062-1. [9] L. Wang and X.-S. Yang, A vast amount of various invariant tori in the Nosé-Hoover oscillator, Chaos, 25 (2015), 123110, 6 pp. doi: 10.1063/1.4937167. [10] L. Wang and X.-S. Yang, The coexistence of invariant tori and topological horseshoe in a generalized Nosé-Hoover oscillator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750111, 12 pp. doi: 10.1142/S0218127417501115. [11] L. Wang and X.-S. Yang, Global analysis of a generalized Nosé-Hoover oscillator, J. Math. Anal. Appl., 464 (2018), 370-379.  doi: 10.1016/j.jmaa.2018.04.013.
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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