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Invariant manifolds for random dynamical systems with slow and fast variables. (English) Zbl 1138.37032
The paper discusses existence of inertial manifolds for a family of random differential equations in \(\mathbb R^{d_1}\times\mathbb R^{d_2}\) with a fast and a slow component, scaled by a factor \(0<\varepsilon\ll1\), where it is assumed that the influence of randomness splits accordingly into a fast and a slow part.

MSC:
37H99 Random dynamical systems
34F05 Ordinary differential equations and systems with randomness
34C25 Periodic solutions to ordinary differential equations
37D10 Invariant manifold theory for dynamical systems
70K70 Systems with slow and fast motions for nonlinear problems in mechanics
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[1] Arnold L. (1998) Random Dynamical Systems. Springer, New York · Zbl 0906.34001
[2] Bensoussan A., Flandoli F. (1995) Stochastic inertial manifold. Stochastics Stochastics Rep. 53(1–2): 13–39 · Zbl 0854.60059
[3] Berglund N., Gentz B. (2003) Geometric singular perturbation theory for stochastic differential equations. J. Diff. Equat. 191, 1–54 · Zbl 1053.34048 · doi:10.1016/S0022-0396(03)00020-2
[4] Butuzov V.F., Kalachev L. V., Vasil’eva A.B. (1995) The Boundary Function Method for Singular Perturbation Problems. SIAM Studies in Applied Mathematics, Philadelphia · Zbl 0823.34059
[5] Castaing, C., Valadier, M. (1977). Convex Analysis and Measurable Multifunctions. Lect. Notes in Math. 580. Springer-Verlag, Berlin, Heidelberg, New York. · Zbl 0346.46038
[6] Da Prato G., Debussche A. (1996) Construction of stochastic inertial manifolds using backward integration. Stochastics Stochastics Rep. 59(3–4): 305–324 · Zbl 0876.60040
[7] Duan J., Lu K., Schmalfuß B. (2003). Invariant manifolds for stochastic partial differential equations. Ann. Probab. 31, 2109–2135 · Zbl 1052.60048 · doi:10.1214/aop/1068646380
[8] Eckhaus W. (1973). Matched Asymptotic Expansions and Singular Perturbations. North-Holland Publishing Comp., Amsterdam · Zbl 0255.34002
[9] Fenichel N. (1971) Persistence and smoothness of invariant manifolds for flows. Math. J., Indiana Univ. 21, 193–226 · Zbl 0246.58015 · doi:10.1512/iumj.1971.21.21017
[10] Fenichel N. (1979) Geometric singular perturbation theory for ordinary differential equations. J. Diff. Equat. 31, 53–98 · Zbl 0476.34034 · doi:10.1016/0022-0396(79)90152-9
[11] Girya T.V., Chueshov I.D. (1995) Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems. Sb. Math. 186(1): 29–45 · Zbl 0851.60036 · doi:10.1070/SM1995v186n01ABEH000002
[12] Hadamard J. (1901) Sur l’iteration et les solutions asymptotiques des equations differenttielles. Bull. Soc. Math. France 29, 224–228 · JFM 32.0314.01
[13] Handrock-Meyer S., Kalachev L.V., Schneider K.R. (2001) A method to determine the dimension of long-time dynamics in multi-scale systems. J. Math. Chem. 30(2): 133–160 · Zbl 0983.92037 · doi:10.1023/A:1017960802671
[14] Kabanov, Y., and Pergamenshchikov, S. (2003). Two–Scale Stochastic Systems. Asymptotic Analysis and Control. Stochstic Modelling and Applied Probability 49. Springer, Berlin. · Zbl 1033.60001
[15] Kevorkian J.K., Cole J.D. (1996) Multiple Scale and Singular Perturbation Methods. Springer-Verlag, New-York, Berlin · Zbl 0846.34001
[16] Koksch N., Siegmund S. (2002) Pullback attracting inertial manifolds for nonautonomous dynamical systems. J. Dyn. Diff. Equat. 14, 889–441 · Zbl 1025.34042 · doi:10.1023/A:1020768711975
[17] Lagerstrom, P. A. (1988). Matched Asymptotic Expansions. Applied Mathematical Sciences 76, Springer-Verlag, New York. · Zbl 0666.34064
[18] Oono Y. (2000) Renormalization and asymptotics. RIMS Kokyuroku 1134, 1–18 · Zbl 0958.82500
[19] Pötzsche C., Siegmund S. (2004). C m -smoothness of invariant fiber bundles. Topol. Meth. Nonlinear Anal. 24(1): 107–145 · Zbl 1075.39014
[20] Sanders J.A., Verhulst F. (1985) Averaging Methods in Nonlinear Dynamical Systems. Springer-Verlag, New York · Zbl 0586.34040
[21] Schmalfuß B. (1998) A random fixed point theorem and the random graph transformation. J. Math. Anal. Appl. 225(1): 91–113 · Zbl 0931.37019 · doi:10.1006/jmaa.1998.6008
[22] Schneider K.R., Wilhelm T. (2000) Model reduction by extended quasi-steady-state approximation. J. Math. Biol. 40, 443–450 · Zbl 0970.92028 · doi:10.1007/s002850000026
[23] Segel L.A. (1989) On the validity of the steady state assumption of enzyme kinetics. SIAM Rev. 31, 446–477 · Zbl 0679.34066 · doi:10.1137/1031091
[24] Strygin V.V., Sobolev V.A. (1988) Separation of Motions by the Method of Integral Manifolds (in Russian). Nauka, Moscow · Zbl 0657.70002
[25] Wiggins S. (1994) Normally Hyperbolic Invariant Manifolds in Dynamical Systems. Springer-Verlag, New York · Zbl 0812.58001
[26] Zeidler E. (1985) Nonlinear Functional Analysis and its Applications, Vol. 1. Springer-Verlag, Berlin, Heidelberg, New York · Zbl 0583.47051
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