American Institute of Mathematical Sciences

Advanced
May  2015, 20(3): 861-874. doi: 10.3934/dcdsb.2015.20.861

On Lyapunov exponents of difference equations with random delay

Nguyen Dinh Cong 1, , Thai Son Doan 2, and  Stefan Siegmund 3,

1. 

Institute of Mathematics, Vietnamese Academy of Science and Technology, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam
2. 

Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Ha Noi, Vietnam
3. 

Institute for Analysis & Center for Dynamics, Department of Mathematics, Technische Universität Dresden, Zellescher Weg 12-14, 01069 Dresden, Germany

Received  November 2013 Revised  August 2014 Published  January 2015

The multiplicative ergodic theorem by Oseledets on Lyapunov spectrum and Oseledets subspaces is extended to linear random difference equations with random delay. In contrast to the general multiplicative ergodic theorem by Lian and Lu, we can prove that a random dynamical system generated by a difference equation with random delay cannot have infinitely many Lyapunov exponents.
Keywords: Random difference equations, random delay, multiplicative ergodic theorem, Lyapunov exponent..
Mathematics Subject Classification: Primary: 37H10, 37H15; Secondary: 39A0.
Citation: Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861
References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

L. Arnold and N. D. Cong, Generic properties of Lyapunov exponents, Random Comput. Dynam., 2 (1994), 335-345.  Google Scholar

[3]

I. P. Cornfeld, S. V. Fomin and Y. G. Sinaĭ, Ergodic Theory, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[4]

H. Crauel, T. S. Doan and S. Siegmund, Difference equations with random delay, J. Difference Equ. Appl., 15 (2009), 627-647. doi: 10.1080/10236190802612865.  Google Scholar

[5]

T. S. Doan and S. Siegmund, Differential equations with random delay, Fields Inst. Commun., 64 (2013), 279-303. doi: 10.1007/978-1-4614-4523-4_11.  Google Scholar

[6]

R. C. Ferreira, M. R. S. Briones and F. Antoneli, A model of gene expression based on random dynamical systems reveals modularity properties of gene regulatory networks, preprint, arXiv:1309.0765, 2013. Google Scholar

[7]

M. J. Garrido-Atienza, A. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388. doi: 10.1142/S0219493711003358.  Google Scholar

[8]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991.  Google Scholar

[9]

Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, Mem. Amer. Math. Soc., 206 (2010), vi+106 pp. doi: 10.1090/S0065-9266-10-00574-0.  Google Scholar

[10]

P. Walter, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[11]

F. Wu, G. G. Yin and L. Y. Wang, Stability of a pure random delay system with two-time-scale Markovian switching, J. Differential Equations, 253 (2012), 878-905. doi: 10.1016/j.jde.2012.04.017.  Google Scholar

[12]

F. Wu and P. E. Kloeden, Mean-square random attractors of stochastic delay differential equations with random delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1715-1734. doi: 10.3934/dcdsb.2013.18.1715.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

L. Arnold and N. D. Cong, Generic properties of Lyapunov exponents, Random Comput. Dynam., 2 (1994), 335-345.  Google Scholar

[3]

I. P. Cornfeld, S. V. Fomin and Y. G. Sinaĭ, Ergodic Theory, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[4]

H. Crauel, T. S. Doan and S. Siegmund, Difference equations with random delay, J. Difference Equ. Appl., 15 (2009), 627-647. doi: 10.1080/10236190802612865.  Google Scholar

[5]

T. S. Doan and S. Siegmund, Differential equations with random delay, Fields Inst. Commun., 64 (2013), 279-303. doi: 10.1007/978-1-4614-4523-4_11.  Google Scholar

[6]

R. C. Ferreira, M. R. S. Briones and F. Antoneli, A model of gene expression based on random dynamical systems reveals modularity properties of gene regulatory networks, preprint, arXiv:1309.0765, 2013. Google Scholar

[7]

M. J. Garrido-Atienza, A. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388. doi: 10.1142/S0219493711003358.  Google Scholar

[8]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991.  Google Scholar

[9]

Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, Mem. Amer. Math. Soc., 206 (2010), vi+106 pp. doi: 10.1090/S0065-9266-10-00574-0.  Google Scholar

[10]

P. Walter, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[11]

F. Wu, G. G. Yin and L. Y. Wang, Stability of a pure random delay system with two-time-scale Markovian switching, J. Differential Equations, 253 (2012), 878-905. doi: 10.1016/j.jde.2012.04.017.  Google Scholar

[12]

F. Wu and P. E. Kloeden, Mean-square random attractors of stochastic delay differential equations with random delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1715-1734. doi: 10.3934/dcdsb.2013.18.1715.  Google Scholar

[1]

Pedro Duarte, Silvius Klein, Manuel Santos. A random cocycle with non Hölder Lyapunov exponent. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4841-4861. doi: 10.3934/dcds.2019197
[2]

Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2235-2255. doi: 10.3934/cpaa.2020098
[3]

Rajeshwari Majumdar, Phanuel Mariano, Hugo Panzo, Lowen Peng, Anthony Sisti. Lyapunov exponent and variance in the CLT for products of random matrices related to random Fibonacci sequences. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4779-4799. doi: 10.3934/dcdsb.2020126
[4]

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
[5]

Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269
[6]

Chi Phan. Random attractor for stochastic Hindmarsh-Rose equations with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3233-3256. doi: 10.3934/dcdsb.2020060
[7]

Xuping Zhang. Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021107
[8]

Fuke Wu, Peter E. Kloeden. Mean-square random attractors of stochastic delay differential equations with random delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715
[9]

Alex Blumenthal. A volume-based approach to the multiplicative ergodic theorem on Banach spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2377-2403. doi: 10.3934/dcds.2016.36.2377
[10]

Luciana A. Alves, Luiz A. B. San Martin. Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1247-1273. doi: 10.3934/dcds.2013.33.1247
[11]

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
[12]

Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757
[13]

Tomás Caraballo, M. J. Garrido-Atienza, B. Schmalfuss, José Valero. Non--autonomous and random attractors for delay random semilinear equations without uniqueness. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 415-443. doi: 10.3934/dcds.2008.21.415
[14]

Yangrong Li, Shuang Yang. Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1155-1175. doi: 10.3934/cpaa.2019056
[15]

James Nolen. A central limit theorem for pulled fronts in a random medium. Networks & Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167
[16]

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
[17]

Tao Jiang, Xianming Liu, Jinqiao Duan. Approximation for random stable manifolds under multiplicative correlated noises. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3163-3174. doi: 10.3934/dcdsb.2016091
[18]

Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875
[19]

Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Principal Floquet subspaces and exponential separations of type Ⅱ with applications to random delay differential equations. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6163-6193. doi: 10.3934/dcds.2018265
[20]

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

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (71)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences