Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation

Previous Article
Global existence and uniqueness of the solution for the fractional Schrödinger-KdV-Burgers system
DCDS-S Home
This Issue
Next Article
Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum

December 2016, 9(6): 1701-1715. doi: 10.3934/dcdss.2016071

Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation

Yanfeng Guo ^1, , Jinqiao Duan ^2, and Donglong Li ^1,

1.	School of Science, Guangxi University of Science and Technology, Liuzhou, Guangxi 545006, China, China
2.	Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616

Received June 2015 Revised September 2016 Published November 2016

Abstract
Related Papers

Random invariant manifolds are considered for a stochastic Swift-Hohenberg equation with multiplicative noise in the Stratonovich sense. Using a stochastic transformation and a technique of cut-off function, existence of random invariant manifolds and attracting property of the corresponding random dynamical system are obtained by Lyaponov-Perron method. Then in the sense of large probability, an approximation of invariant manifolds has been investigated and this is further used to describe the geometric shape of the invariant manifolds.

Keywords: stochastic transformation, Stochastic partial differential equations, approximation of invariant manifolds, Lyaponov-Perron method., random invariant manifolds.

Mathematics Subject Classification: Primary: 60H15, 37H05; Secondary: 37L55, 37L2.

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

References:

[1]	L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998). doi: 10.1007/978-3-662-12878-7.
[2]	D. Blömker, M. Hairer and G. A. Pavliotis, Stochastic Swift-Hohenberg equaion near a change of stability,, Proceedings of Equadiff, 11 (2005), 27.
[3]	D. Blömker and W. Wang, Qualitative properties of local random invariant manifolds for SPDE with quadratic nonlinearity,, J. Dyn. Differ. Equ., 22 (2010), 677. doi: 10.1007/s10884-009-9145-6.
[4]	D. Blömker, Amplitude Equations for Stochastic Partial Differential Equations,, vol.3 of Interdisciplinary Mathematical Sciences. World Scientific Publishing, (2007). doi: 10.1142/9789812770608.
[5]	G. Chen, J. Duan and J. Zhang, Geometric shape of invariant manifolds for a class of stochastic partial differential equations,, J. Math. Phys., 52 (2011). doi: 10.1063/1.3614777.
[6]	M. D. Chekroun, H. H. Liu and S. H. Wang, Stochastic Parameterizing Manifolds and non-Markovian Reduced Equations-Stochastic Manifolds for Nonlinear SPDEs II,, Springer Briefs in Mathematics, (2015). doi: 10.1007/978-3-319-12520-6.
[7]	I. Chueshov, Monotone Random Systems Theory and Applications,, Springer-Verlag, (2002). doi: 10.1007/b83277.
[8]	H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Differ. Equ., 9 (1997), 307. doi: 10.1007/BF02219225.
[9]	H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Rel. Fields, 100 (1994), 365. doi: 10.1007/BF01193705.
[10]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223.
[11]	J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic differential equations,, Ann. Probab., 31 (2003), 2109. doi: 10.1214/aop/1068646380.
[12]	J. Duan and K. Lu, Smooth stable and unstable manifolds for stoahcstic partial differential equations,, J. Dyn. Diff. Equ., 16 (2004), 949. doi: 10.1007/s10884-004-7830-z.
[13]	J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations,, Elsevier, (2014).
[14]	J. P. Eckmann and C. E. Wayne, Propagating fronts and the center manifold theorem,, Commun. Math. Phys, 136 (1991), 285. doi: 10.1007/BF02100026.
[15]	H. Fu, X. Liu and J. Duan, Slow manifolds for multi-time-scale stochastic evolutionary systems,, Commun. Math. Sci., 11 (2013), 141. doi: 10.4310/CMS.2013.v11.n1.a5.
[16]	D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Varlag, (1981).
[17]	D. Y. Hsieh, A. Q. Tang and X. P. Wang, On hydrodynamics instabilities, chaos, and phase transition,, Acta Mech. Sin., 12 (1996), 1. doi: 10.1007/BF02486757.
[18]	M. F. Hilali, S. Metens, P. Borckmans and G. Dewel, Pattern selection in the generalized Swift-Hohenberg equation,, Phys. Rev. E, 51 (1995), 2046.
[19]	P. Imkeller and A. Monahan, Conceptual stochastic climate dynamics,, Stoch. Dynam., 2 (2002), 311.
[20]	G. Lin, H. Gao, J. Duan and V. J. Ervin, Asymptotic dynamical difference between the nonlocal and local Swift-Hohenberg models,, J. Math. Phys., 41 (2000), 2077. doi: 10.1063/1.533228.
[21]	A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains: Existence and comparison,, Nonlinearity, 8 (1995), 734. doi: 10.1088/0951-7715/8/5/006.
[22]	J. Oh, J. M. Ortiz de Zárate, J. V. Sengers and G. Ahlers, Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bnárd convection,, Phys. Rev. E, 69 (2004).
[23]	J. Oh and G. Ahlers, Thermal-noise effect on the transition to Rayleigh-Bnárd convection,, Phys. Rev. Lett., 91 (2003).
[24]	A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations,, Springer-Varlag, (1983). doi: 10.1007/978-1-4612-5561-1.
[25]	I. Rehberg, S. Rasenat, M. de la Torre Juárez, W. Schöpf, F. Hörner, G. Ahlers and H. R. Brand, Thermally induced hydrodynamic fluctuations below the onset of electroconvection,, Phys. Rev. Lett., 67 (1991), 596. doi: 10.1103/PhysRevLett.67.596.
[26]	X. Sun, J. Duan and X. Li, An impact of noise on invariant manifolds in nonlinear dynamical systems,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3371010.
[27]	J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability,, Phys. Rev. A, 15 (1977), 319. doi: 10.1103/PhysRevA.15.319.
[28]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988). doi: 10.1007/978-1-4684-0313-8.
[29]	W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2800164.
[30]	W. Wang, J. Sun and J. Duan, Ergodic dynamics of the stochastic Swift-Hohenberg system,, Nonlinear Analysis: Real World Appl., 6 (2005), 273. doi: 10.1016/j.nonrwa.2004.08.009.
[31]	E. Waymire and J. Duan, Probability and Partial Differential Equations in Modern Applied Mathematics,, IMA, (2005). doi: 10.1007/978-0-387-29371-4.

show all references

References:

[1]	L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998). doi: 10.1007/978-3-662-12878-7.
[2]	D. Blömker, M. Hairer and G. A. Pavliotis, Stochastic Swift-Hohenberg equaion near a change of stability,, Proceedings of Equadiff, 11 (2005), 27.
[3]	D. Blömker and W. Wang, Qualitative properties of local random invariant manifolds for SPDE with quadratic nonlinearity,, J. Dyn. Differ. Equ., 22 (2010), 677. doi: 10.1007/s10884-009-9145-6.
[4]	D. Blömker, Amplitude Equations for Stochastic Partial Differential Equations,, vol.3 of Interdisciplinary Mathematical Sciences. World Scientific Publishing, (2007). doi: 10.1142/9789812770608.
[5]	G. Chen, J. Duan and J. Zhang, Geometric shape of invariant manifolds for a class of stochastic partial differential equations,, J. Math. Phys., 52 (2011). doi: 10.1063/1.3614777.
[6]	M. D. Chekroun, H. H. Liu and S. H. Wang, Stochastic Parameterizing Manifolds and non-Markovian Reduced Equations-Stochastic Manifolds for Nonlinear SPDEs II,, Springer Briefs in Mathematics, (2015). doi: 10.1007/978-3-319-12520-6.
[7]	I. Chueshov, Monotone Random Systems Theory and Applications,, Springer-Verlag, (2002). doi: 10.1007/b83277.
[8]	H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Differ. Equ., 9 (1997), 307. doi: 10.1007/BF02219225.
[9]	H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Rel. Fields, 100 (1994), 365. doi: 10.1007/BF01193705.
[10]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223.
[11]	J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic differential equations,, Ann. Probab., 31 (2003), 2109. doi: 10.1214/aop/1068646380.
[12]	J. Duan and K. Lu, Smooth stable and unstable manifolds for stoahcstic partial differential equations,, J. Dyn. Diff. Equ., 16 (2004), 949. doi: 10.1007/s10884-004-7830-z.
[13]	J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations,, Elsevier, (2014).
[14]	J. P. Eckmann and C. E. Wayne, Propagating fronts and the center manifold theorem,, Commun. Math. Phys, 136 (1991), 285. doi: 10.1007/BF02100026.
[15]	H. Fu, X. Liu and J. Duan, Slow manifolds for multi-time-scale stochastic evolutionary systems,, Commun. Math. Sci., 11 (2013), 141. doi: 10.4310/CMS.2013.v11.n1.a5.
[16]	D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Varlag, (1981).
[17]	D. Y. Hsieh, A. Q. Tang and X. P. Wang, On hydrodynamics instabilities, chaos, and phase transition,, Acta Mech. Sin., 12 (1996), 1. doi: 10.1007/BF02486757.
[18]	M. F. Hilali, S. Metens, P. Borckmans and G. Dewel, Pattern selection in the generalized Swift-Hohenberg equation,, Phys. Rev. E, 51 (1995), 2046.
[19]	P. Imkeller and A. Monahan, Conceptual stochastic climate dynamics,, Stoch. Dynam., 2 (2002), 311.
[20]	G. Lin, H. Gao, J. Duan and V. J. Ervin, Asymptotic dynamical difference between the nonlocal and local Swift-Hohenberg models,, J. Math. Phys., 41 (2000), 2077. doi: 10.1063/1.533228.
[21]	A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains: Existence and comparison,, Nonlinearity, 8 (1995), 734. doi: 10.1088/0951-7715/8/5/006.
[22]	J. Oh, J. M. Ortiz de Zárate, J. V. Sengers and G. Ahlers, Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bnárd convection,, Phys. Rev. E, 69 (2004).
[23]	J. Oh and G. Ahlers, Thermal-noise effect on the transition to Rayleigh-Bnárd convection,, Phys. Rev. Lett., 91 (2003).
[24]	A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations,, Springer-Varlag, (1983). doi: 10.1007/978-1-4612-5561-1.
[25]	I. Rehberg, S. Rasenat, M. de la Torre Juárez, W. Schöpf, F. Hörner, G. Ahlers and H. R. Brand, Thermally induced hydrodynamic fluctuations below the onset of electroconvection,, Phys. Rev. Lett., 67 (1991), 596. doi: 10.1103/PhysRevLett.67.596.
[26]	X. Sun, J. Duan and X. Li, An impact of noise on invariant manifolds in nonlinear dynamical systems,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3371010.
[27]	J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability,, Phys. Rev. A, 15 (1977), 319. doi: 10.1103/PhysRevA.15.319.
[28]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988). doi: 10.1007/978-1-4684-0313-8.
[29]	W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2800164.
[30]	W. Wang, J. Sun and J. Duan, Ergodic dynamics of the stochastic Swift-Hohenberg system,, Nonlinear Analysis: Real World Appl., 6 (2005), 273. doi: 10.1016/j.nonrwa.2004.08.009.
[31]	E. Waymire and J. Duan, Probability and Partial Differential Equations in Modern Applied Mathematics,, IMA, (2005). doi: 10.1007/978-0-387-29371-4.

[1]	Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227
[2]	Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203
[3]	Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579
[4]	Nguyen Thieu Huy, Pham Van Bang. Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2993-3011. doi: 10.3934/dcdsb.2015.20.2993
[5]	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
[6]	Rovella Alvaro, Vilamajó Francesc, Romero Neptalí. Invariant manifolds for delay endomorphisms. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 35-50. doi: 10.3934/dcds.2001.7.35
[7]	Arne Ogrowsky, Björn Schmalfuss. Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1663-1681. doi: 10.3934/dcdsb.2013.18.1663
[8]	Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[9]	Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1
[10]	C. M. Groothedde, J. D. Mireles James. Parameterization method for unstable manifolds of delay differential equations. Journal of Computational Dynamics, 2017, 4 (1&2) : 21-70. doi: 10.3934/jcd.2017002
[11]	María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473
[12]	Lok Ming Lui, Yalin Wang, Tony F. Chan, Paul M. Thompson. Brain anatomical feature detection by solving partial differential equations on general manifolds. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 605-618. doi: 10.3934/dcdsb.2007.7.605
[13]	Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[14]	Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085
[15]	José F. Alves, Davide Azevedo. Statistical properties of diffeomorphisms with weak invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 1-41. doi: 10.3934/dcds.2016.36.1
[16]	George Osipenko. Indestructibility of invariant locally non-unique manifolds. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 203-219. doi: 10.3934/dcds.1996.2.203
[17]	Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967
[18]	Arturo Echeverría-Enríquez, Alberto Ibort, Miguel C. Muñoz-Lecanda, Narciso Román-Roy. Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds. Journal of Geometric Mechanics, 2012, 4 (4) : 397-419. doi: 10.3934/jgm.2012.4.397
[19]	Pablo Aguirre, Bernd Krauskopf, Hinke M. Osinga. Global invariant manifolds near a Shilnikov homoclinic bifurcation. Journal of Computational Dynamics, 2014, 1 (1) : 1-38. doi: 10.3934/jcd.2014.1.1
[20]	Roberto Castelli. Efficient representation of invariant manifolds of periodic orbits in the CRTBP. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 563-586. doi: 10.3934/dcdsb.2018197

2017 Impact Factor: 0.561

American Institute of Mathematical Sciences