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

Invariant foliations for random dynamical systems

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.
Citation: Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639
References:
[1]

L. M. Arnold, Random Dynamical Systems,, Springer, (1998). Google Scholar

[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). doi: 10.1090/memo/0645. Google Scholar

[3]

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

[4]

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

[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. Google Scholar

[6]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions,, LNM 580. Springer-Verlag, (1977). Google Scholar

[7]

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

[8]

P. Drabek and J. Milota, Methods of Nonlinear Analysis Applications to Differential Equations,, Birkhäuser Verlag, (2007). Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes in Mathematics, 583 (1977). Google Scholar

[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. doi: 10.1006/jdeq.1994.1025. Google Scholar

[15]

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

[16]

P-D. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems,, Lecture Notes in Mathematics, (1606). Google Scholar

[17]

R. Mañé, Liapunov exponents and stable manifolds for compact transformations, Geometrical dynamics,, Lecture Notes in Math., 1007 (1983), 522. doi: 10.1007/BFb0061433. Google Scholar

[18]

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

[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). doi: 10.1090/S0065-9266-10-00574-0. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

T. Wanner, Linearization of random dynamical systems,, Dynamics Reported, 4 (1995), 203. Google Scholar

[24]

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

show all references

References:
[1]

L. M. Arnold, Random Dynamical Systems,, Springer, (1998). Google Scholar

[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). doi: 10.1090/memo/0645. Google Scholar

[3]

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

[4]

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

[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. Google Scholar

[6]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions,, LNM 580. Springer-Verlag, (1977). Google Scholar

[7]

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

[8]

P. Drabek and J. Milota, Methods of Nonlinear Analysis Applications to Differential Equations,, Birkhäuser Verlag, (2007). Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes in Mathematics, 583 (1977). Google Scholar

[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. doi: 10.1006/jdeq.1994.1025. Google Scholar

[15]

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

[16]

P-D. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems,, Lecture Notes in Mathematics, (1606). Google Scholar

[17]

R. Mañé, Liapunov exponents and stable manifolds for compact transformations, Geometrical dynamics,, Lecture Notes in Math., 1007 (1983), 522. doi: 10.1007/BFb0061433. Google Scholar

[18]

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

[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). doi: 10.1090/S0065-9266-10-00574-0. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

T. Wanner, Linearization of random dynamical systems,, Dynamics Reported, 4 (1995), 203. Google Scholar

[24]

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

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