American Institute of Mathematical Sciences

Advanced
February  2005, 12(2): 355-362. doi: 10.3934/dcds.2005.12.355

Shadowing in random dynamical systems

Lianfa He 1, , Hongwen Zheng 2, and  Yujun Zhu 1,

1. 

College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050016, China, China
2. 

Institute of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China

Received  August 2003 Revised  August 2004 Published  December 2004

In this paper we consider the shadowing property for $C^1$ random dynamical systems. We first define a type of hyperbolicity on the full measure invariant set which is given by Oseledec's multiplicative ergodic theorem, and then prove that the system has the "Lipschitz" shadowing property on it.
Keywords: multiplicative ergodic theorem, shadowing property., Random dynamical system.
Mathematics Subject Classification: 37C50, 37H9.
Citation: 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
[1]

Cecilia González-Tokman, Anthony Quas. A concise proof of the multiplicative ergodic theorem on Banach spaces. Journal of Modern Dynamics, 2015, 9: 237-255. doi: 10.3934/jmd.2015.9.237
[2]

Jonathan Meddaugh. Shadowing as a structural property of the space of dynamical systems. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2439-2451. doi: 10.3934/dcds.2021197
[3]

Alex Blumenthal. A volume-based approach to the multiplicative ergodic theorem on Banach spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2377-2403. doi: 10.3934/dcds.2016.36.2377
[4]

Luciana A. Alves, Luiz A. B. San Martin. Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1247-1273. doi: 10.3934/dcds.2013.33.1247
[5]

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
[6]

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
[7]

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
[8]

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
[9]

Manseob Lee, Jumi Oh, Xiao Wen. Diffeomorphisms with a generalized Lipschitz shadowing property. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1913-1927. doi: 10.3934/dcds.2020346
[10]

Fang Zhang, Yunhua Zhou. On the limit quasi-shadowing property. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2861-2879. doi: 10.3934/dcds.2017123
[11]

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
[12]

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
[13]

Jihoon Lee, Ngocthach Nguyen. Flows with the weak two-sided limit shadowing property. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4375-4395. doi: 10.3934/dcds.2021040
[14]

Shrey Sanadhya. A shrinking target theorem for ergodic transformations of the unit interval. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4003-4011. doi: 10.3934/dcds.2022042
[15]

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
[16]

Mariko Arisawa, Hitoshi Ishii. Some properties of ergodic attractors for controlled dynamical systems. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 43-54. doi: 10.3934/dcds.1998.4.43
[17]

Xiaoyue Li, Xuerong Mao. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 523-545. doi: 10.3934/dcds.2009.24.523
[18]

Noriaki Kawaguchi. Topological stability and shadowing of zero-dimensional dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2743-2761. doi: 10.3934/dcds.2019115
[19]

Yuri Kifer. Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2687-2716. doi: 10.3934/dcds.2018113
[20]

Tao Jiang, Xianming Liu, Jinqiao Duan. Approximation for random stable manifolds under multiplicative correlated noises. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3163-3174. doi: 10.3934/dcdsb.2016091

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (131)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy