# American Institute of Mathematical Sciences

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. 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.

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##### 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.
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 $PI$
The lower part of $B$. The dotted curve is the intersection of $S$ with $P$
(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
(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
(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"
(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
(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
(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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