American Institute of Mathematical Sciences

Advanced
March  2007, 18(2&3): 597-611. doi: 10.3934/dcds.2007.18.597

Global attractor and rotation number of a class of nonlinear noisy oscillators

Wenxian Shen 1,

1. 

Department of Mathematics and Statistics, Auburn University, Auburn University, AL 36849-5310, United States

Received  March 2006 Revised  August 2006 Published  March 2007

The current paper is concerned with the global dynamics of a class of nonlinear oscillators driven by real or white noises, of which a typical example is a shunted Josephson junction exposed to some random medium. Applying random dynamical systems theory, it is shown that a driven oscillator in the class under consideration with a tempered real noise has a one-dimensional global random attractor provided that the damping is not too small. Moreover, restricted to the global attractor, the oscillator induces a random dynamical system on $S^1$. It is then shown that there is a rotation number associated to the oscillator which characterizes the speed at which the solutions of the oscillator move around the global attractor. The results extend the existing ones for time periodic and quasi-periodic Josephson junctions and can be applied to Josephson junctions driven by white noises.
Keywords: tempered function, global random attractor, invariant measure, ergodic theorem, random fixed point, real or white noise, random dynamical system, rotation number., Josephson junction.
Mathematics Subject Classification: 34C15, 34F05, 34D45,37H10, 60H1.
Citation: Wenxian Shen. Global attractor and rotation number of a class of nonlinear noisy oscillators. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 597-611. doi: 10.3934/dcds.2007.18.597
[1]

Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757
[2]

Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210
[3]

Xingni Tan, Fuqi Yin, Guihong Fan. Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3153-3170. doi: 10.3934/dcdsb.2020055
[4]

Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887
[5]

Ling Xu, Jianhua Huang, Qiaozhen Ma. Random exponential attractor for stochastic non-autonomous suspension bridge equation with additive white noise. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021318
[6]

Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093
[7]

Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639
[8]

Min Zhao, Shengfan Zhou. Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1683-1717. doi: 10.3934/dcdsb.2017081
[9]

Xiangnan He, Wenlian Lu, Tianping Chen. On transverse stability of random dynamical system. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 701-721. doi: 10.3934/dcds.2013.33.701
[10]

Weigu Li, Kening Lu. A Siegel theorem for dynamical systems under random perturbations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 635-642. doi: 10.3934/dcdsb.2008.9.635
[11]

Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285
[12]

Fawwaz Batayneh, Cecilia González-Tokman. On the number of invariant measures for random expanding maps in higher dimensions. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5887-5914. doi: 10.3934/dcds.2021100
[13]

Chi Phan. Random attractor for stochastic Hindmarsh-Rose equations with multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3233-3256. doi: 10.3934/dcdsb.2020060
[14]

Jun Shen, Kening Lu, Bixiang Wang. Invariant manifolds and foliations for random differential equations driven by colored noise. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6201-6246. doi: 10.3934/dcds.2020276
[15]

Julian Newman. Synchronisation of almost all trajectories of a random dynamical system. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4163-4177. doi: 10.3934/dcds.2020176
[16]

N. D. Cong, T. S. Doan, S. Siegmund. A Bohl-Perron type theorem for random dynamical systems. Conference Publications, 2011, 2011 (Special) : 322-331. doi: 10.3934/proc.2011.2011.322
[17]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355
[18]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1
[19]

Sijia Zhang, Shengfan Zhou. Random uniform exponential attractors for Schrödinger lattice systems with quasi-periodic forces and multiplicative white noise. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022056
[20]

Robert Hesse, Alexandra Neamţu. Global solutions and random dynamical systems for rough evolution equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2723-2748. doi: 10.3934/dcdsb.2020029

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy