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.

show all references

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.

[1]

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

[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 and Applied Analysis, 2020, 19 (4) : 2235-2255. doi: 10.3934/cpaa.2020098

[3]

Fumihiko Nakamura, Yushi Nakano, Hisayoshi Toyokawa. Lyapunov exponents for random maps. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022058

[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]

Dimitri Breda, Sara Della Schiava. Pseudospectral reduction to compute Lyapunov exponents of delay differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2727-2741. doi: 10.3934/dcdsb.2018092

[6]

Nguyen Dinh Cong, Nguyen Thi Thuy Quynh. Coincidence of Lyapunov exponents and central exponents of linear Ito stochastic differential equations with nondegenerate stochastic term. Conference Publications, 2011, 2011 (Special) : 332-342. doi: 10.3934/proc.2011.2011.332

[7]

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

[8]

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

[9]

Yuri Bakhtin. Lyapunov exponents for stochastic differential equations with infinite memory and application to stochastic Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 697-709. doi: 10.3934/dcdsb.2006.6.697

[10]

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

[11]

Florian Dorsch, Hermann Schulz-Baldes. Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 945-976. doi: 10.3934/dcdsb.2021076

[12]

Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216

[13]

Igor G. Vladimirov. The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 575-600. doi: 10.3934/dcdsb.2013.18.575

[14]

Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure and Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565

[15]

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

[16]

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

[17]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91

[18]

Edson de Faria, Pablo Guarino. Real bounds and Lyapunov exponents. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1957-1982. doi: 10.3934/dcds.2016.36.1957

[19]

Zoltán Buczolich, Gabriella Keszthelyi. Isentropes and Lyapunov exponents. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 1989-2009. doi: 10.3934/dcds.2020102

[20]

Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (136)
  • HTML views (1)
  • Cited by (1)

Other articles
by authors

[Back to Top]