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Non-autonomous stochastic evolution equations of parabolic type with nonlocal initial conditions
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 |
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 $.
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. 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. |
[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. |
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. |
[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. 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. |
[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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