-
Previous Article
Optimal control strategies for an online game addiction model with low and high risk exposure
- DCDS-B Home
- This Issue
-
Next Article
Singular support of the global attractor for a damped BBM equation
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 |
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 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. |
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.
|
| [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. |
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
| [1] |
Qianqian Han, Bo Deng, Xiao-Song Yang. The existence of $ \omega $-limit set for a modified Nosé-Hoover oscillator. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022043 |
| [2] |
Alain Léger, Elaine Pratt. On the equilibria and qualitative dynamics of a forced nonlinear oscillator with contact and friction. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 501-527. doi: 10.3934/dcdss.2016009 |
| [3] |
Jorge A. Esquivel-Avila. Qualitative analysis of a nonlinear wave equation. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 787-804. doi: 10.3934/dcds.2004.10.787 |
| [4] |
W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209 |
| [5] |
Aeeman Fatima, F. M. Mahomed, Chaudry Masood Khalique. Conditional symmetries of nonlinear third-order ordinary differential equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 655-666. doi: 10.3934/dcdss.2018040 |
| [6] |
Alessandro Fonda, Fabio Zanolin. Bounded solutions of nonlinear second order ordinary differential equations. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 91-98. doi: 10.3934/dcds.1998.4.91 |
| [7] |
Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3589-3610. doi: 10.3934/dcdss.2021021 |
| [8] |
Daniel G. Alfaro Vigo, Amaury C. Álvarez, Grigori Chapiro, Galina C. García, Carlos G. Moreira. Solving the inverse problem for an ordinary differential equation using conjugation. Journal of Computational Dynamics, 2020, 7 (2) : 183-208. doi: 10.3934/jcd.2020008 |
| [9] |
Richard A. Norton, G. R. W. Quispel. Discrete gradient methods for preserving a first integral of an ordinary differential equation. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1147-1170. doi: 10.3934/dcds.2014.34.1147 |
| [10] |
Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268 |
| [11] |
Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709 |
| [12] |
Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203 |
| [13] |
Kiah Wah Ong. Dynamic transitions of generalized Kuramoto-Sivashinsky equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1225-1236. doi: 10.3934/dcdsb.2016.21.1225 |
| [14] |
Jordi-Lluís Figueras, Àlex Haro. A note on the fractalization of saddle invariant curves in quasiperiodic systems. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1095-1107. doi: 10.3934/dcdss.2016043 |
| [15] |
L. Bakker. A reducible representation of the generalized symmetry group of a quasiperiodic flow. Conference Publications, 2003, 2003 (Special) : 68-77. doi: 10.3934/proc.2003.2003.68 |
| [16] |
Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807 |
| [17] |
Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299 |
| [18] |
Yanfei Lu, Qingfei Yin, Hongyi Li, Hongli Sun, Yunlei Yang, Muzhou Hou. Solving higher order nonlinear ordinary differential equations with least squares support vector machines. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1481-1502. doi: 10.3934/jimo.2019012 |
| [19] |
Lijuan Wang, Yashan Xu. Admissible controls and controllable sets for a linear time-varying ordinary differential equation. Mathematical Control and Related Fields, 2018, 8 (3&4) : 1001-1019. doi: 10.3934/mcrf.2018043 |
| [20] |
Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure and Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]