Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.
Readers can access Online First articles via the “Online First” tab for the selected journal.
| 1. | School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou, Henan 450046, China |
| 2. | Department of mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA |
| 3. | School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China |
| 4. | Hubei Key Laboratory of Engineering Modeling and Science Computing, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China |
In this paper, we prove the existence of $ \omega $-limit set for a modified Nosé-Hoover oscillator. We also prove the existence of either an invariant torus or a stable periodic orbit of the oscillator. In addition, we show by numerical simulations the co-existence of both $ \alpha $- and $ \omega $-limit sets of various types of periodic orbits, invariant tori, and chaotic attractors.
| [1] |
C. Chicone, Ordinary Differential Equations with Applications, 2$^{nd}$ edition, Springer, New York, 2006. |
| [2] |
W. G. Hoover,
Canonical dynamics: Equilibrium phase-space distributions, Phys. Rev. A, 31 (1985), 1695-1697.
doi: 10.1103/PhysRevA.31.1695. |
| [3] |
W. G. Hoover,
Remark on "Some simple chaotic flows", Physical Review E, 51 (1995), 759-759.
|
| [4] |
W. G. Hoover,
Nosé–Hoover nonequilibrium dynamics and statistical mechanics, Molecular Simulation, 33 (2007), 13-19.
|
| [5] |
S. Nosé,
A unified formulation of the constant temperature molecular dynamics methods, Journal of Chemical Physics, 81 (1984), 511-519.
doi: 10.1143/PTPS.103.1. |
| [6] |
S. Nosé,
A molecular dynamics method for simulations in the canonical ensemble, Molecular Physics, 52 (2002), 255-268.
|
| [7] |
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. |
| [8] |
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, Physical Review E, 89 (2014), 042914-042914.
doi: 10.1103/PhysRevE.89.042914. |
| [9] |
W. Tucker,
A rigorous ODE solver and smale's 14th problem, Found. Comput. Math., 2 (2002), 53-117.
doi: 10.1007/s002080010018. |
| [10] |
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. |
| [11] |
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. |
show all references
| [1] |
C. Chicone, Ordinary Differential Equations with Applications, 2$^{nd}$ edition, Springer, New York, 2006. |
| [2] |
W. G. Hoover,
Canonical dynamics: Equilibrium phase-space distributions, Phys. Rev. A, 31 (1985), 1695-1697.
doi: 10.1103/PhysRevA.31.1695. |
| [3] |
W. G. Hoover,
Remark on "Some simple chaotic flows", Physical Review E, 51 (1995), 759-759.
|
| [4] |
W. G. Hoover,
Nosé–Hoover nonequilibrium dynamics and statistical mechanics, Molecular Simulation, 33 (2007), 13-19.
|
| [5] |
S. Nosé,
A unified formulation of the constant temperature molecular dynamics methods, Journal of Chemical Physics, 81 (1984), 511-519.
doi: 10.1143/PTPS.103.1. |
| [6] |
S. Nosé,
A molecular dynamics method for simulations in the canonical ensemble, Molecular Physics, 52 (2002), 255-268.
|
| [7] |
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. |
| [8] |
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, Physical Review E, 89 (2014), 042914-042914.
doi: 10.1103/PhysRevE.89.042914. |
| [9] |
W. Tucker,
A rigorous ODE solver and smale's 14th problem, Found. Comput. Math., 2 (2002), 53-117.
doi: 10.1007/s002080010018. |
| [10] |
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. |
| [11] |
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. |
| [1] |
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 |
| [2] |
Harry Dankowicz, Jan Sieber. Sensitivity analysis for periodic orbits and quasiperiodic invariant tori using the adjoint method. Journal of Computational Dynamics, 2022, 9 (3) : 329-369. doi: 10.3934/jcd.2022006 |
| [3] |
Changjing Zhuge, Xiaojuan Sun, Jinzhi Lei. On positive solutions and the Omega limit set for a class of delay differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2487-2503. doi: 10.3934/dcdsb.2013.18.2487 |
| [4] |
Tifei Qian, Zhihong Xia. Heteroclinic orbits and chaotic invariant sets for monotone twist maps. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 69-95. doi: 10.3934/dcds.2003.9.69 |
| [5] |
Roberto Castelli. Efficient representation of invariant manifolds of periodic orbits in the CRTBP. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 563-586. doi: 10.3934/dcdsb.2018197 |
| [6] |
Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185 |
| [7] |
Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098 |
| [8] |
Lorenzo Arona, Josep J. Masdemont. Computation of heteroclinic orbits between normally hyperbolic invariant 3-spheres foliated by 2-dimensional invariant Tori in Hill's problem. Conference Publications, 2007, 2007 (Special) : 64-74. doi: 10.3934/proc.2007.2007.64 |
| [9] |
Victoriano Carmona, Emilio Freire, Soledad Fernández-García. Periodic orbits and invariant cones in three-dimensional piecewise linear systems. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 59-72. doi: 10.3934/dcds.2015.35.59 |
| [10] |
Alexei Pokrovskii, Oleg Rasskazov, Daniela Visetti. Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator. Discrete and Continuous Dynamical Systems - B, 2007, 8 (4) : 943-970. doi: 10.3934/dcdsb.2007.8.943 |
| [11] |
Anatoly Neishtadt, Carles Simó, Dmitry Treschev, Alexei Vasiliev. Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 621-650. doi: 10.3934/dcdsb.2008.10.621 |
| [12] |
Tiago de Carvalho, Rodrigo Donizete Euzébio, Jaume Llibre, Durval José Tonon. Detecting periodic orbits in some 3D chaotic quadratic polynomial differential systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 1-11. doi: 10.3934/dcdsb.2016.21.1 |
| [13] |
Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569 |
| [14] |
Àlex Haro, Rafael de la Llave. A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1261-1300. doi: 10.3934/dcdsb.2006.6.1261 |
| [15] |
Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671 |
| [16] |
Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439 |
| [17] |
Andrew D. Barwell, Chris Good, Piotr Oprocha, Brian E. Raines. Characterizations of $\omega$-limit sets in topologically hyperbolic systems. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1819-1833. doi: 10.3934/dcds.2013.33.1819 |
| [18] |
Hongyong Cui, Peter E. Kloeden, Meihua Yang. Forward omega limit sets of nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1103-1114. doi: 10.3934/dcdss.2020065 |
| [19] |
Bruce Kitchens, Michał Misiurewicz. Omega-limit sets for spiral maps. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 787-798. doi: 10.3934/dcds.2010.27.787 |
| [20] |
Shengqing Hu, Bin Liu. Degenerate lower dimensional invariant tori in reversible system. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3735-3763. doi: 10.3934/dcds.2018162 |
2021 Impact Factor: 1.497
Article outline
Figures and Tables
[Back to Top]
