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 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.

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