# American Institute of Mathematical Sciences

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

## On Lyapunov exponents of difference equations with random delay

 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.
Citation: Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete and 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. [2] L. Arnold and N. D. Cong, Generic properties of Lyapunov exponents, Random Comput. Dynam., 2 (1994), 335-345. [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. [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. [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. [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. [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. [8] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991. [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. [10] P. Walter, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. [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. [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.

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##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. [2] L. Arnold and N. D. Cong, Generic properties of Lyapunov exponents, Random Comput. Dynam., 2 (1994), 335-345. [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. [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. [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. [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. [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. [8] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991. [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. [10] P. Walter, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. [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. [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.
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