doi: 10.3934/dcdsb.2022043
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The existence of $ \omega $-limit set for a modified Nosé-Hoover oscillator

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

*Corresponding author: Xiao-Song Yang

Received  November 2021 Revised  January 2022 Early access March 2022

Fund Project: The first author is supported by NSF grant 51979116

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.

Citation: 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, doi: 10.3934/dcdsb.2022043
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. PoschW. 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. SprottW. 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

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. PoschW. 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. SprottW. 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.

Figure 1.  The upper part of $ B $. The downward parabola look-like is the intersection of the $ z $-nullcline $ P $ with the cylinder $ V(x, y) = M^2 $. The dotted line is the intersection of the $ y $-nullcline and the $ z $-nullcline. On the $ F_1 $ face, the vector field points into $ PE $ and on the $ F_2 $ face, it points into $ PI $
Figure 2.  The lower part of $ B $. The dotted curve is the intersection of $ S $ with $ P $
Figure 3.  (a)-(f) are invariant tori with $ T_{p} $ = 1:2, 2:3, 3:4, 2.9999:3.9999, 5.0002:6.0003, 7.0007:8.0007 and initial values $ X_{0} $ = (0.0776, 1.9386, 1.9512), (-0.2889, 0.4881, -0.9277), (0.8234, 2.3778, -1.2032), (0.7020, -2.2847, 0.1837), (1.6250, 2.3172, 1.2231), (2.3138, 1.8492, 1.8285) respectively; (g) and (h) are periodic orbits with $ T_{p} $ = 4.9999:5.9999, 9.0007:10.0008 and initial values $ X_{0} $ = (-0.7839, 0.4982, 2.9191), (3.3447, -0.0000, -0.0082) respectively; (i) is a chaotic orbit with initial value $ X_{0} = (0.9209, -0.1560, 0.9179) $ and Lyapunov index Ly = 0.3859
Figure 4.  (a)-(i) are the intersection of the orbits in Fig. 3 with $ \Sigma $ respectively, the transformation of the color from cool to warm represents the forward passage in time for these orbits
Figure 5.  (a)-(i) are the orbits with the same initial values as in Fig. 3. But (a)-(h) are with $ T_{p} $ = 3.0248:4.0245, 4.6841:5.6841, 3.0001:4.0001, 2.9999:3.9999, 5.0003:6.0003, 7.0001:8.0002, 4.9999:5.9999, 9.0007:10.0008 respectively, (i) is a "chaotic attractor"
Figure 6.  (a)-(i) are the intersection of the orbits in Fig. 5 with $ \Sigma $ respectively, the usage of colors in this figure are the same as that in Fig. 4
Figure 7.  (a)-(c), (d)-(f) are the $ \alpha $-sets and $ \omega $-sets of the orbits with the initial values (0.0757, -1.2747, 2.4021), (0.3868, -0.7685, -0.9078) and (0.3522, 2.4609, 1.7899) respectively; (g)-(i) are the coexistence of these limit sets. Orbits in green are for $ \alpha $-sets and those in purple are for $ \omega $-sets. For (g), the $ \alpha $-set is "hidden" mostly inside the $ \omega $-set, and vice versa for (h). For (i), neither set is blocked out from view by the other
Figure 8.  (a)-(c) are the intersections of the orbits in Fig. 7 with $ \Sigma $ respectively, the transition of the color from blue to green represents the backward passage in time, and that from cyan to purple represents the forward passage in time for these orbits
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