American Institute of Mathematical Sciences

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doi: 10.3934/dcdsb.2020346

Qualitative analysis of a generalized Nosé-Hoover oscillator

Qianqian Han 1, and  Xiao-Song Yang 2,3,,

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  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 $.

Keywords: Quasiperiodic orbit, qualitative dynamic, generalized Nosé-Hoover oscillator, invariant set, nonlinear ordinary differential equation.
Mathematics Subject Classification: Primary: 34C15, 34C27; Secondary: 37C55.
Citation: Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 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.  Google Scholar

[2]

W. G. Hoover, Canonical dynamics: Equilibrium phase-space distributions, Phys. Rev. A, 31 (1985), 1695-1697.  doi: 10.1103/PhysRevA.31.1695.  Google Scholar

[3]

S. Nosé, A unified formulation of the constant temperature molecular dynamics methods, Journal of Chemical Physics, 81 (1984), 511-519.   Google Scholar

[4]

S. Nosé, A molecular dynamics method for simulations in the canonical ensemble, Molecular Physics, 52 (2002), 255-268.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

W. G. Hoover, Canonical dynamics: Equilibrium phase-space distributions, Phys. Rev. A, 31 (1985), 1695-1697.  doi: 10.1103/PhysRevA.31.1695.  Google Scholar

[3]

S. Nosé, A unified formulation of the constant temperature molecular dynamics methods, Journal of Chemical Physics, 81 (1984), 511-519.   Google Scholar

[4]

S. Nosé, A molecular dynamics method for simulations in the canonical ensemble, Molecular Physics, 52 (2002), 255-268.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The grid is part of $ S_{1} $ and the shadow is part of $ S_{2} $
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Figure 2.  The shadow is the projection of the region $ I $ on the plane $ z = 0 $
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Figure 3.  $ 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
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Figure 4.  From right to left are $ l_{10} $ and $ l_{20} $
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Figure 5.  From right to left are $ l_{01} $, $ l_{02} $, $ l_{03} $ and $ l_{04} $
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