American Institute of Mathematical Sciences

Advanced
September  2014, 34(9): 3639-3666. doi: 10.3934/dcds.2014.34.3639

Invariant foliations for random dynamical systems

Ji Li 1, , Kening Lu 2, and  Peter W. Bates 3,

1. 

Institute for Mathematics and its Application, University of Minnesota, Minneapolis, MN, 55455, United States
2. 

Department of Mathematics, Brigham Young University, Provo, Utah 84602
3. 

Department of Mathematics, Michigan State University, East Lansing, MI 48824

Received  June 2013 Revised  December 2013 Published  March 2014

We prove the existence of invariant foliations of stable and unstable manifolds of a normally hyperbolic random invariant manifold. The normally hyperbolic random invariant manifold referred to here is that which was shown to exist in a previous paper when a deterministic dynamical system having a normally hyperbolic invariant manifold is subjected to a small random perturbation.
Keywords: random normally hyperbolic invariant manifolds, Random dynamical systems, random invariant foliations..
Mathematics Subject Classification: Primary: 34C37, 34C45, 34F05, 37H1.
Citation: Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639
References:
[1]

L. M. Arnold, Random Dynamical Systems, Springer, New York, 1998.

[2]

P. Bates, K. Lu and C. Zeng, Existence and Persistence of Invariant Manifolds for Semiflows in Banach Space, Memoirs of the AMS, 135 1998, viii+129 pp. doi: 10.1090/memo/0645.

[3]

P. Bates, K. Lu and C. Zeng, Persistence of overflowing manifold for semiflow, Comm. Pure Appl. Math., 52 (1999), 983-1046. doi: 10.1002/(SICI)1097-0312(199908)52:8<983::AID-CPA4>3.0.CO;2-O.

[4]

P. Bates, K. Lu and C. Zeng, Invariant foliations near normally hyperbolic invariant manifolds for semiflows, Trans. Amer. Math. Soc., 352 (2000), 4641-4676. doi: 10.1090/S0002-9947-00-02503-4.

[5]

T. Caraballo, J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for random and stochastic partial differential equations, Adv. Nonlinear Stud., 10 (2010), 23-52.

[6]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580. Springer-Verlag, Berlin-Heidelberg-New York, 1977.

[7]

S-N. Chow, K. Lu and X-B. Lin, Smooth foliations for flows in banach space, Journal of Differential Equations, 94 (1991), 266-291. doi: 10.1016/0022-0396(91)90093-O.

[8]

P. Drabek and J. Milota, Methods of Nonlinear Analysis Applications to Differential Equations, Birkhäuser Verlag, Basel, 2007.

[9]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. Journal, 21 (1971), 193-226. doi: 10.1512/iumj.1972.21.21017.

[10]

N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. Journal, 23 (1974), 1109-1137.

[11]

N. Fenichel, Asymptotic stability with rate conditions II, Indiana Univ. Math. Journal, 26 (1977), 81-93. doi: 10.1512/iumj.1977.26.26006.

[12]

J. Hadamard, Sur l'iteration et les solutions asymptotiques des equations defferentielles, Bull. Soc. Math. France, 29 (1901), 224-228.

[13]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, New York, 1977.

[14]

C. K. R. T. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations, 108 (1994), 64-88. doi: 10.1006/jdeq.1994.1025.

[15]

J. Li, K. Lu and P. Bates, Normally hyperbolic invariant manifolds for random dynamical systems, Trans. Amer. Math. Soc., 365 (2013), 5933-5966. doi: 10.1090/S0002-9947-2013-05825-4.

[16]

P-D. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems, Lecture Notes in Mathematics, 1606. Springer-Verlag, Berlin, 1995.

[17]

R. Mañé, Liapunov exponents and stable manifolds for compact transformations, Geometrical dynamics, Lecture Notes in Math., Springer Verlag, New York, 1007 (1983), 522-577. doi: 10.1007/BFb0061433.

[18]

W. Li and K. Lu, Sternberg theorems for random dynamical systems, Comm. Pure Appl. Math., 58 (2005), 941-988. doi: 10.1002/cpa.20083.

[19]

Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, Memoirs of the AMS., 206 (2010), vi+106 pp. doi: 10.1090/S0065-9266-10-00574-0.

[20]

K. Lu and B. Schmalfuss, Invariant foliations for stochastic partial differential equations, Stoch. Dyn., 8 (2008), 505-518. doi: 10.1142/S0219493708002421.

[21]

Y. Pesin, Characteristic Lyapunov exponents, and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-112.

[22]

D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert spaces, Ann. of Math., 115 (1982), 243-290. doi: 10.2307/1971392.

[23]

T. Wanner, Linearization of random dynamical systems, Dynamics Reported,, Springer-Verlag, New York, 4 (1995), 203-269.

[24]

H. Whitney, Differential manifolds, Ann. of Math., 37 (1936), 645-680. doi: 10.2307/1968482.

show all references

References:
[1]

L. M. Arnold, Random Dynamical Systems, Springer, New York, 1998.

[2]

P. Bates, K. Lu and C. Zeng, Existence and Persistence of Invariant Manifolds for Semiflows in Banach Space, Memoirs of the AMS, 135 1998, viii+129 pp. doi: 10.1090/memo/0645.

[3]

P. Bates, K. Lu and C. Zeng, Persistence of overflowing manifold for semiflow, Comm. Pure Appl. Math., 52 (1999), 983-1046. doi: 10.1002/(SICI)1097-0312(199908)52:8<983::AID-CPA4>3.0.CO;2-O.

[4]

P. Bates, K. Lu and C. Zeng, Invariant foliations near normally hyperbolic invariant manifolds for semiflows, Trans. Amer. Math. Soc., 352 (2000), 4641-4676. doi: 10.1090/S0002-9947-00-02503-4.

[5]

T. Caraballo, J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for random and stochastic partial differential equations, Adv. Nonlinear Stud., 10 (2010), 23-52.

[6]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580. Springer-Verlag, Berlin-Heidelberg-New York, 1977.

[7]

S-N. Chow, K. Lu and X-B. Lin, Smooth foliations for flows in banach space, Journal of Differential Equations, 94 (1991), 266-291. doi: 10.1016/0022-0396(91)90093-O.

[8]

P. Drabek and J. Milota, Methods of Nonlinear Analysis Applications to Differential Equations, Birkhäuser Verlag, Basel, 2007.

[9]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. Journal, 21 (1971), 193-226. doi: 10.1512/iumj.1972.21.21017.

[10]

N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. Journal, 23 (1974), 1109-1137.

[11]

N. Fenichel, Asymptotic stability with rate conditions II, Indiana Univ. Math. Journal, 26 (1977), 81-93. doi: 10.1512/iumj.1977.26.26006.

[12]

J. Hadamard, Sur l'iteration et les solutions asymptotiques des equations defferentielles, Bull. Soc. Math. France, 29 (1901), 224-228.

[13]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, New York, 1977.

[14]

C. K. R. T. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations, 108 (1994), 64-88. doi: 10.1006/jdeq.1994.1025.

[15]

J. Li, K. Lu and P. Bates, Normally hyperbolic invariant manifolds for random dynamical systems, Trans. Amer. Math. Soc., 365 (2013), 5933-5966. doi: 10.1090/S0002-9947-2013-05825-4.

[16]

P-D. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems, Lecture Notes in Mathematics, 1606. Springer-Verlag, Berlin, 1995.

[17]

R. Mañé, Liapunov exponents and stable manifolds for compact transformations, Geometrical dynamics, Lecture Notes in Math., Springer Verlag, New York, 1007 (1983), 522-577. doi: 10.1007/BFb0061433.

[18]

W. Li and K. Lu, Sternberg theorems for random dynamical systems, Comm. Pure Appl. Math., 58 (2005), 941-988. doi: 10.1002/cpa.20083.

[19]

Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, Memoirs of the AMS., 206 (2010), vi+106 pp. doi: 10.1090/S0065-9266-10-00574-0.

[20]

K. Lu and B. Schmalfuss, Invariant foliations for stochastic partial differential equations, Stoch. Dyn., 8 (2008), 505-518. doi: 10.1142/S0219493708002421.

[21]

Y. Pesin, Characteristic Lyapunov exponents, and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-112.

[22]

D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert spaces, Ann. of Math., 115 (1982), 243-290. doi: 10.2307/1971392.

[23]

T. Wanner, Linearization of random dynamical systems, Dynamics Reported,, Springer-Verlag, New York, 4 (1995), 203-269.

[24]

H. Whitney, Differential manifolds, Ann. of Math., 37 (1936), 645-680. doi: 10.2307/1968482.

[1]

Jun Shen, Kening Lu, Bixiang Wang. Invariant manifolds and foliations for random differential equations driven by colored noise. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6201-6246. doi: 10.3934/dcds.2020276
[2]

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

Henk Broer, Aaron Hagen, Gert Vegter. Numerical approximation of normally hyperbolic invariant manifolds. Conference Publications, 2003, 2003 (Special) : 133-140. doi: 10.3934/proc.2003.2003.133
[4]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324
[5]

Yanfeng Guo, Jinqiao Duan, Donglong Li. Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1701-1715. doi: 10.3934/dcdss.2016071
[6]

Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123
[7]

Maciej J. Capiński, Piotr Zgliczyński. Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 641-670. doi: 10.3934/dcds.2011.30.641
[8]

Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089
[9]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355
[10]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1
[11]

Yujun Zhu. Preimage entropy for random dynamical systems. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 829-851. doi: 10.3934/dcds.2007.18.829
[12]

Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093
[13]

Fawwaz Batayneh, Cecilia González-Tokman. On the number of invariant measures for random expanding maps in higher dimensions. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5887-5914. doi: 10.3934/dcds.2021100
[14]

Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 727-744. doi: 10.3934/dcds.2003.9.727
[15]

Weigu Li, Kening Lu. A Siegel theorem for dynamical systems under random perturbations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 635-642. doi: 10.3934/dcdsb.2008.9.635
[16]

Yuri Kifer. Computations in dynamical systems via random perturbations. Discrete and Continuous Dynamical Systems, 1997, 3 (4) : 457-476. doi: 10.3934/dcds.1997.3.457
[17]

Thomas Bogenschütz, Achim Doebler. Large deviations in expanding random dynamical systems. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 805-812. doi: 10.3934/dcds.1999.5.805
[18]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166
[19]

Xinsheng Wang, Weisheng Wu, Yujun Zhu. Local unstable entropy and local unstable pressure for random partially hyperbolic dynamical systems. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 81-105. doi: 10.3934/dcds.2020004
[20]

Maciej J. Capiński. Covering relations and the existence of topologically normally hyperbolic invariant sets. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 705-725. doi: 10.3934/dcds.2009.23.705

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (186)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy