Advanced Search
Article Contents
Article Contents

Invariant manifolds and foliations for random differential equations driven by colored noise

This work was supported by NSFC (11501549, 11831012, 11331007, 11971330), NSF (1413603), the Fundamental Research Funds for the Central Universities (YJ201646) and International Visiting Program for Excellent Young Scholars of SCU

Abstract Full Text(HTML) Related Papers Cited by
  • In this paper, we prove the existence of local stable and unstable invariant manifolds for a class of random differential equations driven by nonlinear colored noise defined in a fractional power of a separable Banach space. In the case of linear noise, we show the pathwise convergence of these random invariant manifolds as well as invariant foliations as the correlation time of the colored noise approaches zero.

    Mathematics Subject Classification: Primary: 60H15; Secondary: 37H10, 37L55, 37D10.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
    [2] P. W. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Rep., 2 (1989), 1-38. 
    [3] P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in banach space, Mem. Amer. Math. Soc., 135 (1998), 129 pp. doi: 10.1090/memo/0645.
    [4] P. W. BatesK. 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] P. W. BatesK. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states, Invent. math., 174 (2008), 355-433.  doi: 10.1007/s00222-008-0141-y.
    [6] T. CaraballoP. E. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.
    [7] J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York-Berlin, 1981.
    [8] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580. Springer-Verlag, Berlin-New York, 1977.
    [9] S-N. ChowX.-B. Lin and K. Lu, Smooth invariant foliations in infinite dimensional spaces, J. Differential Equations, 94 (1991), 266-291.  doi: 10.1016/0022-0396(91)90093-O.
    [10] S.-N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74 (1988), 285-317.  doi: 10.1016/0022-0396(88)90007-1.
    [11] S.-N. Chow and K. Lu, $C^k$ center unstable manifolds, Proc. Roy. Soc. Edinburgh Sect. A, 108 (1988), 303-320.  doi: 10.1017/S0308210500014682.
    [12] G. Da Prato and  J. Zabczyk.Stochastic Equations in Infinite Dimension., Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781107295513.
    [13] J. DuanK. Lu and B. Schmalfuß, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380.
    [14] J. DuanK. Lu and B. Schmalfuß, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972.  doi: 10.1007/s10884-004-7830-z.
    [15] F. Flandoli, Regularity Theory and Stochastic Flow for Parabolic SPDEs, Stochastics Monographs Vol. 9, Gordon and Breach Science Publishers, Yverdon, 1995.
    [16] H. GaoM. J. Garrido-Atienza and B. Schmalfuß, Random attractors for stochastic evolution equations driven by fractional Brownian motion, SIAM J. Math. Anal., 46 (2014), 2281-2309.  doi: 10.1137/130930662.
    [17] M. J. Garrido-AtienzaK. Lu and B. Schmalfuß, Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion, J. Differential Equations, 248 (2010), 1637-1667.  doi: 10.1016/j.jde.2009.11.006.
    [18] M. J. Garrido-AtienzaK. Lu and B. Schmalfuß, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H\in(1/3, 1/2]$, SIAM J. Appl. Dyn. Syst., 15 (2016), 625-654.  doi: 10.1137/15M1030303.
    [19] A. Gu and B. Wang, Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Disc. Cont. Dyn. Sys.-Series B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.
    [20] J. Hadamard, Surl'iteration et les solutions asymptotiquesd es equations differentielles, Bull. Soc. Math. France, 29 (1901), 224-228. 
    [21] J. K. Hale, Ordinary Differential Equations, Wiley, New York, 1969.
    [22] D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, New York, 1981.
    [23] T. JiangX. Liu and J. Duan, Approximation for random stable manifolds under multiplicative correlated noises, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3163-3174.  doi: 10.3934/dcdsb.2016091.
    [24] T. Jiang, X. Liu and J. Duan, A Wong-Zakai approximation for random invariant manifolds, J. Math. Phys., 58 (2017), 122701, 19pp. doi: 10.1063/1.5017932.
    [25] A. Kelley, The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations, 3 (1967), 546-570.  doi: 10.1016/0022-0396(67)90016-2.
    [26] A. Liapounoff, ProblÜme Gén´eral de la Stabilité du Mouvement, Princeton Univ. Press, Princeton, N. J., 1947.
    [27] K. Lu and B. Schmalfuß, Invariant foliations for stochastic partial differential equations, Stoch. Dyn., 8 (2008), 505-518.  doi: 10.1142/S0219493708002421.
    [28] S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for stochastic differential equations, Ann. Probab., 27 (1999), 615-652.  doi: 10.1214/aop/1022677380.
    [29] S.-E. A. Mohammed, T. Zhang and H. Zhao, The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations, Memories AMS Vol. 196, Amer. Math. Soc., Providence, R.I., 2008. doi: 10.1090/memo/0917.
    [30] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
    [31] O. Perron, $\ddot{\text{U}}$ber Stabilit$\ddot{\text{a}}$t und asymptotisches Verhalten der Integrale von Differentialgleichungssysteme, Math. Z., 29 (1928), 129-160.  doi: 10.1007/BF01180524.
    [32] V. A. Pliss, A reduction principle in the theory of stability of motion, Izv. Akad. Nauk SSSR Ser. Mat., 28 (1964), 1297–1324 (in Russian).
    [33] D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math., 115 (1982), 243-290.  doi: 10.2307/1971392.
    [34] B. Schmalfuß, A random fixed point theorem and the random graph transformation, J. Math. Anal. Appl., 225 (1998), 91-113.  doi: 10.1006/jmaa.1998.6008.
    [35] J. Shen and K. Lu, Wong-Zakai approximations and center manifolds of stochastic differential equations, J. Differential Equations, 263 (2017), 4929-4977.  doi: 10.1016/j.jde.2017.06.005.
    [36] J. Shen, K. Lu and B. Wang, Convergence and center manifolds for differential equations driven by colored noise, Discrete Contin. Dyn. Syst. Ser. A, 39 (2019), 4797-4840. doi: 10.3934/dcds.2019196.
    [37] J. Shen, J. Zhao, K. Lu and B. Wang, The Wong-Zakai approximations of invariant manifolds and foliations for stochastic evolution equations, J. Differential Equations, 266 (2019), 4568-4623. doi: 10.1016/j.jde.2018.10.008.
    [38] A. Vanderbauwhede and S. A. van Gils, Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal., 72 (1987), 209-224.  doi: 10.1016/0022-1236(87)90086-3.
    [39] T. Wanner, Linearization of random dynamical systems, in: C. K. R. T. Jonesm, U. Kirchgraber, H. O. Walther (Eds.), Dynamics Rep., vol. 4,203–269, Springer, Berlin/Heidelberg/New York, 1995. doi: 10.1007/978-3-642-61215-2_4.
  • 加载中

Article Metrics

HTML views(379) PDF downloads(342) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint