# American Institute of Mathematical Sciences

August  2016, 9(4): 995-1007. doi: 10.3934/dcdss.2016038

## On integral separation of bounded linear random differential equations

 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 Road, Ha Noi, Viet Nam, Vietnam

Received  August 2015 Revised  January 2016 Published  August 2016

Our aim in this paper is to investigate the openness and denseness for the set of integrally separated systems in the space of bounded linear random differential equations equipped with the $L^{\infty}$-metric. We show that in the general case, the set of integrally separated systems is open and dense. An exception is the case when the base space is isomorphic to the ergodic rotation flow of the unit circle, in which the set of integrally separated systems is open but not dense.
Citation: Nguyen Dinh Cong, Doan Thai Son. On integral separation of bounded linear random differential equations. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 995-1007. doi: 10.3934/dcdss.2016038
##### References:
 [1] L. Ya. Adrianova, Introduction to Linear Systems of Differential Equations, Translations of Mathematical Monographs, 146. American Mathematical Society, Providence, RI, 1995. [2] W. Ambrose, Representation of ergodic flows, Annals of Mathematics, 42 (1941), 723-739. doi: 10.2307/1969259. [3] A. Arbieto and J. Bochi, $L^p$-generic cocycles have one-point Lyapunov spectrum, Stoch. Dynam., 3 (2003), 73-81. (Corrigendum Stoch. Dynam. 3 (2003), 419-420.) doi: 10.1142/S0219493703000619. [4] L. Arnold, Random Dynamical Systems, Springer, 1998. doi: 10.1007/978-3-662-12878-7. [5] L. Arnold and N. D. Cong, Linear cocycles with simple Lyapunov spectrum are dense in $L^{\infty}$, Ergodic Theory Dynam. Systems, 19 (1999), 1389-1404. doi: 10.1017/S014338579915199X. [6] A. Avila, J. Bochi and D. Damanik, Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts, Duke Math. J., 146 (2009), 253-280. doi: 10.1215/00127094-2008-065. [7] M. Bessa, Dynamics of generic $2$-dimensional linear differential systems, J. Diff. Equations, 228 (2006), 685-706. doi: 10.1016/j.jde.2006.03.009. [8] M. Bessa, Dynamic of generic multidimensional linear differenatil systems, Adv. Nonlinear Stud., 8 (2008), 191-211. [9] M. Bessa and H. Vilarinho, Fine properties of $L^p$-cocycles which allow abundance of simple and trivial spectrum, J. Diff. Equations, 256 (2014), 2337-2367. doi: 10.1016/j.jde.2014.01.003. [10] J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and sympletic systems, Ann. Math., 161 (2005), 1423-1485. doi: 10.4007/annals.2005.161.1423. [11] N. D. Cong, A generic bounded linear cocycle has simple Lyapunov spectrum, Ergodic Theory Dynam. Systems, 25 (2005), 1775-1797. doi: 10.1017/S0143385705000337. [12] N. D. Cong and T. S. Doan, An open set of unbounded cocycles with simple Lyapunov spectrum and no exponential separation, Stoch. Dyn., 7 (2007), 335-355. doi: 10.1142/S0219493707002062. [13] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinaĭ, Ergodic Theory, Grundlehren der Mathematischen Wissenschaften, 245. Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5. [14] R. Fabbri and R. Johnson, On the Lyapounov exponent of certain $\mbox{SL}(2,\mathbbR)$-valued cocycles, Differential Equations Dynam. Systems, 7 (1999), 349-370. [15] R. Fabbri and R. Johnson, Genericity of exponential dichotomy for two-dimensional differential systems, Ann. Mat. Pura Appl., 178 (2000), 175-193. doi: 10.1007/BF02505894. [16] R. Fabbri, R. Johnson and R. Pavani, On the nature of the spectrum of the quasi-periodic Schrödinger operator, Nonlinear Anal. Real World Appl., 3 (2002), 37-59. doi: 10.1016/S1468-1218(01)00012-8. [17] R. Fabbri, R. Johnson and L. Zampogni, On the Lyapunov exponent of certain $\mbox{SL}(2,\mathbbR)$-valued cocycles II, Differ. Equ. Dyn. Syst., 18 (2010), 135-161. doi: 10.1007/s12591-010-0003-0. [18] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1976. [19] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176. American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176. [20] V. M. Millionshchikov, Systems with integral separateness which are dense in the set of all linear systems of differential equations, Diff. Equations, 5 (1969), 1167-1170. [21] K. J. Palmer, Exponential separation, exponential dichotomy and spectral theory for linear systems of ordinary differential equations, J. Diff. Equations, 46 (1982), 324-345. doi: 10.1016/0022-0396(82)90098-5. [22] R. J. Sacker and G. R. Sell, A spectral theory for linear differential equations, J. Diff. Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8. [23] S. Siegmund, Dichotomy spectrum for nonautonomous differential equations, J. Dynam. Differential Equations, 14 (2002), 243-258. doi: 10.1023/A:1012919512399.

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##### References:
 [1] L. Ya. Adrianova, Introduction to Linear Systems of Differential Equations, Translations of Mathematical Monographs, 146. American Mathematical Society, Providence, RI, 1995. [2] W. Ambrose, Representation of ergodic flows, Annals of Mathematics, 42 (1941), 723-739. doi: 10.2307/1969259. [3] A. Arbieto and J. Bochi, $L^p$-generic cocycles have one-point Lyapunov spectrum, Stoch. Dynam., 3 (2003), 73-81. (Corrigendum Stoch. Dynam. 3 (2003), 419-420.) doi: 10.1142/S0219493703000619. [4] L. Arnold, Random Dynamical Systems, Springer, 1998. doi: 10.1007/978-3-662-12878-7. [5] L. Arnold and N. D. Cong, Linear cocycles with simple Lyapunov spectrum are dense in $L^{\infty}$, Ergodic Theory Dynam. Systems, 19 (1999), 1389-1404. doi: 10.1017/S014338579915199X. [6] A. Avila, J. Bochi and D. Damanik, Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts, Duke Math. J., 146 (2009), 253-280. doi: 10.1215/00127094-2008-065. [7] M. Bessa, Dynamics of generic $2$-dimensional linear differential systems, J. Diff. Equations, 228 (2006), 685-706. doi: 10.1016/j.jde.2006.03.009. [8] M. Bessa, Dynamic of generic multidimensional linear differenatil systems, Adv. Nonlinear Stud., 8 (2008), 191-211. [9] M. Bessa and H. Vilarinho, Fine properties of $L^p$-cocycles which allow abundance of simple and trivial spectrum, J. Diff. Equations, 256 (2014), 2337-2367. doi: 10.1016/j.jde.2014.01.003. [10] J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and sympletic systems, Ann. Math., 161 (2005), 1423-1485. doi: 10.4007/annals.2005.161.1423. [11] N. D. Cong, A generic bounded linear cocycle has simple Lyapunov spectrum, Ergodic Theory Dynam. Systems, 25 (2005), 1775-1797. doi: 10.1017/S0143385705000337. [12] N. D. Cong and T. S. Doan, An open set of unbounded cocycles with simple Lyapunov spectrum and no exponential separation, Stoch. Dyn., 7 (2007), 335-355. doi: 10.1142/S0219493707002062. [13] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinaĭ, Ergodic Theory, Grundlehren der Mathematischen Wissenschaften, 245. Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5. [14] R. Fabbri and R. Johnson, On the Lyapounov exponent of certain $\mbox{SL}(2,\mathbbR)$-valued cocycles, Differential Equations Dynam. Systems, 7 (1999), 349-370. [15] R. Fabbri and R. Johnson, Genericity of exponential dichotomy for two-dimensional differential systems, Ann. Mat. Pura Appl., 178 (2000), 175-193. doi: 10.1007/BF02505894. [16] R. Fabbri, R. Johnson and R. Pavani, On the nature of the spectrum of the quasi-periodic Schrödinger operator, Nonlinear Anal. Real World Appl., 3 (2002), 37-59. doi: 10.1016/S1468-1218(01)00012-8. [17] R. Fabbri, R. Johnson and L. Zampogni, On the Lyapunov exponent of certain $\mbox{SL}(2,\mathbbR)$-valued cocycles II, Differ. Equ. Dyn. Syst., 18 (2010), 135-161. doi: 10.1007/s12591-010-0003-0. [18] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1976. [19] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176. American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176. [20] V. M. Millionshchikov, Systems with integral separateness which are dense in the set of all linear systems of differential equations, Diff. Equations, 5 (1969), 1167-1170. [21] K. J. Palmer, Exponential separation, exponential dichotomy and spectral theory for linear systems of ordinary differential equations, J. Diff. Equations, 46 (1982), 324-345. doi: 10.1016/0022-0396(82)90098-5. [22] R. J. Sacker and G. R. Sell, A spectral theory for linear differential equations, J. Diff. Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8. [23] S. Siegmund, Dichotomy spectrum for nonautonomous differential equations, J. Dynam. Differential Equations, 14 (2002), 243-258. doi: 10.1023/A:1012919512399.
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