Normally hyperbolic invariant manifolds for random dynamical systems: Part I - persistence

Li, Ji; Lu, Kening; Bates, Peter

doi:10.1090/S0002-9947-2013-05825-4

Normally hyperbolic invariant manifolds for random dynamical systems: Part I - persistence
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by Ji Li, Kening Lu and Peter Bates PDF

Trans. Amer. Math. Soc. 365 (2013), 5933-5966 Request permission

Abstract:

In this paper, we prove the persistence of smooth normally hyperbolic invariant manifolds for dynamical systems under random perturbations.

References

Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992, DOI 10.1007/978-3-662-12878-7
P. W. Bates, P. C. Fife, R. A. Gardner, and C. K. R. T. Jones, Phase field models for hypercooled solidification, Phys. D 104 (1997), no. 1, 1–31. MR 1447948, DOI 10.1016/S0167-2789(96)00207-2
Peter W. Bates, Kening Lu, and Chongchun Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc. 135 (1998), no. 645, viii+129. MR 1445489, DOI 10.1090/memo/0645
Peter W. Bates, Kening Lu, and Chongchun Zeng, Persistence of overflowing manifolds for semiflow, Comm. Pure Appl. Math. 52 (1999), no. 8, 983–1046. MR 1686965, DOI 10.1002/(SICI)1097-0312(199908)52:8<983::AID-CPA4>3.3.CO;2-F
Peter W. Bates, Kening Lu, and Chongchun Zeng, Invariant foliations near normally hyperbolic invariant manifolds for semiflows, Trans. Amer. Math. Soc. 352 (2000), no. 10, 4641–4676. MR 1675237, DOI 10.1090/S0002-9947-00-02503-4
Alain Bensoussan and Franco Flandoli, Stochastic inertial manifold, Stochastics Stochastics Rep. 53 (1995), no. 1-2, 13–39. MR 1380488, DOI 10.1080/17442509508833981
Petra Boxler, A stochastic version of center manifold theory, Probab. Theory Related Fields 83 (1989), no. 4, 509–545. MR 1022628, DOI 10.1007/BF01845701
M. Brin and Yu. Kifer, Dynamics of Markov chains and stable manifolds for random diffeomorphisms, Ergodic Theory Dynam. Systems 7 (1987), no. 3, 351–374. MR 912374, DOI 10.1017/S0143385700004107
P. Brunovský, Tracking invariant manifolds without differential forms, Acta Math. Univ. Comenian. (N.S.) 65 (1996), no. 1, 23–32. MR 1422292
Tomás Caraballo, Jinqiao Duan, Kening Lu, and Björn Schmalfuß, Invariant manifolds for random and stochastic partial differential equations, Adv. Nonlinear Stud. 10 (2010), no. 1, 23–52. MR 2574373, DOI 10.1515/ans-2010-0102
Tomás Caraballo, José A. Langa, and James C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2013, 2041–2061. MR 1857922, DOI 10.1098/rspa.2001.0819
Andrew Carverhill, A formula for the Lyapunov numbers of a stochastic flow. Application to a perturbation theorem, Stochastics 14 (1985), no. 3, 209–226. MR 800244, DOI 10.1080/17442508508833339
C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310, DOI 10.1007/BFb0087685
Giuseppe Da Prato and Arnaud Debussche, Construction of stochastic inertial manifolds using backward integration, Stochastics Stochastics Rep. 59 (1996), no. 3-4, 305–324. MR 1427743, DOI 10.1080/17442509608834094
Stephan Dahlke, Invariant manifolds for products of random diffeomorphisms, J. Dynam. Differential Equations 9 (1997), no. 2, 157–210. MR 1451289, DOI 10.1007/BF02219220
Pavel Drábek and Jaroslav Milota, Methods of nonlinear analysis, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007. Applications to differential equations. MR 2323436
Jinqiao Duan, Kening Lu, and Björn Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab. 31 (2003), no. 4, 2109–2135. MR 2016614, DOI 10.1214/aop/1068646380
Jinqiao Duan, Kening Lu, and Björn Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations 16 (2004), no. 4, 949–972. MR 2110052, DOI 10.1007/s10884-004-7830-z
Neil Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971/72), 193–226. MR 287106, DOI 10.1512/iumj.1971.21.21017
Neil Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J. 23 (1973/74), 1109–1137. MR 339276, DOI 10.1512/iumj.1974.23.23090
Neil Fenichel, Asymptotic stability with rate conditions. II, Indiana Univ. Math. J. 26 (1977), no. 1, 81–93. MR 426056, DOI 10.1512/iumj.1977.26.26006
María J. Garrido-Atienza, Kening Lu, and Björn Schmalfuß, Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion, J. Differential Equations 248 (2010), no. 7, 1637–1667. MR 2593602, DOI 10.1016/j.jde.2009.11.006
T. V. Girya and I. D. Chueshov, Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems, Mat. Sb. 186 (1995), no. 1, 29–46 (Russian, with Russian summary); English transl., Sb. Math. 186 (1995), no. 1, 29–45. MR 1641664, DOI 10.1070/SM1995v186n01ABEH000002
J. Hadamard, Sur l’iteration et les solutions asymptotiques des equations differentielles, Bull. Soc. Math. France 29 (1901), 224-228.
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173, DOI 10.1007/BFb0092042
C. K. R. T. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations 108 (1994), no. 1, 64–88. MR 1268351, DOI 10.1006/jdeq.1994.1025
Christopher K. R. T. Jones, Stability of the travelling wave solution of the FitzHugh-Nagumo system, Trans. Amer. Math. Soc. 286 (1984), no. 2, 431–469. MR 760971, DOI 10.1090/S0002-9947-1984-0760971-6
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Martin Krupa, Björn Sandstede, and Peter Szmolyan, Fast and slow waves in the FitzHugh-Nagumo equation, J. Differential Equations 133 (1997), no. 1, 49–97. MR 1426757, DOI 10.1006/jdeq.1996.3198
Weigu Li and Kening Lu, Sternberg theorems for random dynamical systems, Comm. Pure Appl. Math. 58 (2005), no. 7, 941–988. MR 2142880, DOI 10.1002/cpa.20083
Zeng Lian and Kening Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, Mem. Amer. Math. Soc. 206 (2010), no. 967, vi+106. MR 2674952, DOI 10.1090/S0065-9266-10-00574-0
Pei-Dong Liu and Min Qian, Smooth ergodic theory of random dynamical systems, Lecture Notes in Mathematics, vol. 1606, Springer-Verlag, Berlin, 1995. MR 1369243, DOI 10.1007/BFb0094308
Kening Lu and Björn Schmalfuß, Invariant manifolds for stochastic wave equations, J. Differential Equations 236 (2007), no. 2, 460–492. MR 2322020, DOI 10.1016/j.jde.2006.09.024
Salah-Eldin A. Mohammed and Michael K. R. Scheutzow, The stable manifold theorem for stochastic differential equations, Ann. Probab. 27 (1999), no. 2, 615–652. MR 1698943, DOI 10.1214/aop/1022677380
Salah-Eldin A. Mohammed, Tusheng Zhang, and Huaizhong Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Mem. Amer. Math. Soc. 196 (2008), no. 917, vi+105. MR 2459571, DOI 10.1090/memo/0917
David Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2) 115 (1982), no. 2, 243–290. MR 647807, DOI 10.2307/1971392
Stephen Schecter, Existence of Dafermos profiles for singular shocks, J. Differential Equations 205 (2004), no. 1, 185–210. MR 2094383, DOI 10.1016/j.jde.2004.06.013
Stephen Schecter, Exchange lemmas. II. General exchange lemma, J. Differential Equations 245 (2008), no. 2, 411–441. MR 2428005, DOI 10.1016/j.jde.2007.10.021
Björn Schmalfuss, A random fixed point theorem and the random graph transformation, J. Math. Anal. Appl. 225 (1998), no. 1, 91–113. MR 1639297, DOI 10.1006/jmaa.1998.6008
Wei Wang and Jinqiao Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys. 48 (2007), no. 10, 102701, 14. MR 2362785, DOI 10.1063/1.2800164
Thomas Wanner, Linearization of random dynamical systems, Dynamics reported, Dynam. Report. Expositions Dynam. Systems (N.S.), vol. 4, Springer, Berlin, 1995, pp. 203–269. MR 1346499
Hassler Whitney, Differentiable manifolds, Ann. of Math. (2) 37 (1936), no. 3, 645–680. MR 1503303, DOI 10.2307/1968482