American Institute of Mathematical Sciences

Advanced
October  1997, 3(4): 457-476. doi: 10.3934/dcds.1997.3.457

Computations in dynamical systems via random perturbations

Yuri Kifer 1,

1. 

Institute of Mathematics, Hebrew University, Jerusalem 91904

Received  March 1997 Published  July 1997

I consider discretized random perturbations of hyperbolic dynamical systems and prove that when perturbation parameter tends to zero invariant measures of corresponding Markov chains converge to the Sinai-Bowen-Ruelle measure of the dynamical system. This provides a robust method for computations of such measures and for visualizations of some hyperbolic attractors by modeling randomly perturbed dynamical systems on a computer. Similar results are true for discretized random perturbations of maps of the interval satisfying the Misiurewicz condition considered in [KK].
Keywords: Dynamical systems, random perturbations., discretizations.
Mathematics Subject Classification: Primary: 58F15; Secondary: 34F05, 65L9.
Citation: Yuri Kifer. Computations in dynamical systems via random perturbations. Discrete & Continuous Dynamical Systems, 1997, 3 (4) : 457-476. doi: 10.3934/dcds.1997.3.457
[1]

Weigu Li, Kening Lu. A Siegel theorem for dynamical systems under random perturbations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 635-642. doi: 10.3934/dcdsb.2008.9.635
[2]

Meng Liu, Ke Wang. Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2495-2522. doi: 10.3934/dcds.2013.33.2495
[3]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355
[4]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1
[5]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351
[6]

Jin Zhang, Yonghai Wang, Chengkui Zhong. Robustness of exponentially κ-dissipative dynamical systems with perturbations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3875-3890. doi: 10.3934/dcdsb.2017198
[7]

Yujun Zhu. Preimage entropy for random dynamical systems. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 829-851. doi: 10.3934/dcds.2007.18.829
[8]

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

Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093
[10]

Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 727-744. doi: 10.3934/dcds.2003.9.727
[11]

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

Thomas Bogenschütz, Achim Doebler. Large deviations in expanding random dynamical systems. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 805-812. doi: 10.3934/dcds.1999.5.805
[13]

Jianjun Paul Tian. Finite-time perturbations of dynamical systems and applications to tumor therapy. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 469-479. doi: 10.3934/dcdsb.2009.12.469
[14]

Doan Thai Son. On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3113-3126. doi: 10.3934/dcdsb.2017166
[15]

T. Jäger. Neuronal coding of pacemaker neurons -- A random dynamical systems approach. Communications on Pure & Applied Analysis, 2011, 10 (3) : 995-1009. doi: 10.3934/cpaa.2011.10.995
[16]

Anhui Gu. Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5737-5767. doi: 10.3934/dcdsb.2019104
[17]

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
[18]

Xianfeng Ma, Ercai Chen. Pre-image variational principle for bundle random dynamical systems. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 957-972. doi: 10.3934/dcds.2009.23.957
[19]

Kening Lu, Alexandra Neamţu, Björn Schmalfuss. On the Oseledets-splitting for infinite-dimensional random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1219-1242. doi: 10.3934/dcdsb.2018149
[20]

Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4341-4366. doi: 10.3934/dcdsb.2019122

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (82)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences