随机FitzHugh-Nagumo格点系统的随机吸引子
|
摘要
本文旨在考虑了具有可乘白噪声的FitzHugh-Nagumo格点系统的随机吸引子。通过Ornstein-Uhlenbeck变换,将系统化为随机变量作为系数的随机微分方程,其解确定一个随机动力系统。我们首先证明该随机动力系统的吸收集的存在性,然后证明随机动力系统是渐近紧的,最后得出该系统存在随机吸引子。
The present paper is devoted to the existence of the random attractor of stochastic FitzHugh-Nagumo infinite lattice system with multiplicative white noise. Using the Ornstein-Uhlenbeck transform, the lattice system is transformed into a random differential equation with random coefficients ,which generate a Random Dynamical System. We firstly show the existence of a random bounded absorbing set. Then we prove that the random dynamical system is asymptotically compact. Finally, we prove the existence of the random attractor.
The present paper is devoted to the existence of the random attractor of stochastic FitzHugh-Nagumo infinite lattice system with multiplicative white noise. Using the Ornstein-Uhlenbeck transform, the lattice system is transformed into a random differential equation with random coefficients ,which generate a Random Dynamical System. We firstly show the existence of a random bounded absorbing set. Then we prove that the random dynamical system is asymptotically compact. Finally, we prove the existence of the random attractor.
引文
[1]文兰,动力系统简介,数学进展,31(4)(2002), 293-294.
[2] Birkhoff G D,Dynamical systems,AMS:Publications Providence.
[3]赵才地,非牛顿流与格点系统的渐近行为.上海大学,博士论文
[4] K.Ito On stochastic differential equations.Mem.Amer.Math.Soc.1951 4:51
[5] K.D.Elworthy,Stochastic dynamical systems and their ?ows.In A.Fried-man andM.Pinsky editors Stochastic Analysis Academic Press New York 1978 79 95.
[6] P.H.Baxendale Wiener processes on manifolds of maps.Proc.Roy.Soc.Edinburgh Sect.l980 87A:127 152.
[7] J.M.Bismut A generalized formula of Ito and some other properties of stochastic ?ows.Z.Wahrsch.Verve.Gebiete 1981 55:331 350.
[8] E.Pardoux and S.Peng Adapted solution of a backward stochastic differential equation.Systems ControlLett.1990 14:55 61.
[9] T.Caraballo et al.Upper semicontinuity of attractors for small random perbutations of dynamical syatems.Comm Partieal Differential Equations 1998 23:1557 1581.
[10] H.Crauel and F.Flandoli Attactors for random dynamical systems.Probab.Theory Relat.Fields 1994,100:365 393.
[11] Crauel H.etal.Random attractors.J.Differential Equations 1995 9:307 341.
[12] S.Zhou Attraetors for second order dissipative lattiee dynamical systems J.Differential Equa-tions 2002 179:605 624.
[13] P.W.Bares H.LiseiandK.Lu Attractors for stochastic lattice dynamical systems.Stochastic and Dynam-ics 2006 6:l 21
[14] PW.Bates K.Lu and B.Wang Attractors for lattice dynamical systems.Internationl Joumal of Bifurca-tionand Chaos 200l 11:143 153.
[15] C.D.Zhao and S.F.Zhou Sufficient conditions for the existence of global random attraetors for stochasticlattice dynamical systems and applieations.J.Math.Anal.Appl 2009 l:78 95.
[16] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998.
[17]张锦炎,常微分方程几何理论与分支问题,北京:北京大学出版社,1987.
[18] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on un-bounded domains, J. Differential Equations, 246(2009), 845-869.
[19] A. R. Bishop, K. Fesser, P. S. Lomdahl, and S. E. Trullinger, In?uence of solitons in the initial state onchaos in the driven damped sine-Gordon system, Phys. D 7 (1983), 259–279.
[20] A. R. Bishop and P. S. Lomdahl, Nonlinear dynamics in driven, damped sine-Gordon systems, Phys. D 18(1986), 54–66.
[21] A. N. Borodin, P. Salminen, Handbook of Brownian motion—facts and formulae. Second edition. Proba-bility and its Applications. Birkha¨user Verlag, Basel, 2002.
[22] T. Caraballo, P. E. Kloeden, and J. Real, Pullback and forward attractors for a damped wave equation withdelays, Stoch. Dyn. 4 (2004), 405–423.
[23] S.-N. Chow, W. Shen and H. M. Zhou, Dynamical order in systems of coupled noisy oscillators, J. Dyn.Diff. Eqns. 19(2007), 1007-1035.
[24] S.-N. Chow, X.-B. Lin and K. Lu, Smooth invariant foliations in infinite dimensional spaces, J. Diff. Eq.94(1991), 266-291.
[25] I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, 2002.
[26] H. Crauel, Random Probability Measure on Polish Spaces, Taylor & Francis:London, 2002.
[27] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, 1992.
[28] R. W. Dickey, Stability theory for the damped sine-Gordon equation, SIAM J. Appl. Math. 30 (1976),248–262.
[29] J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann.Probab. 31(2003), 2109 2135.
[30] X. M. Fan, Random attractor for a damped sine-Gordon equation with white noise, Pacific J. Math.216(2004), 63-76.
[31] X. M. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multi-plicative noise, Stoch. Anal. Appl. 24 (2006), 767–793.
[32] J. M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. PuresAppl. 66 (1987), 273-319.
[33] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI,1988.
[34] G. Kovacˇicˇand S. Wiggins, Orbits homoclinic to resonances, with an application to chaos in a model ofthe forced and damped sine-Gordon equation, Phys. D 57 (1992), 185–225
[35] M. Levi, F. C. Hoppensyeadt and W. L. Miranker, Dynamics of the Josephson junction, Quart. Appl.Math.7(1978), 167-198.
[36] R. Mansuy, M. Yor, Aspects of Brownian motion. Universitext. Springer-Verlag, Berlin, 2008.
[37] L. Nana, T. C. Kofan and E. Kaptouom, Subharmonic and homoclinic bifurcations in the driven anddamped sine-Gordon system, Phys. D 134 (1999), 61–74.
[38] E. Nelson Dynamical theories of Brownian motion. Princeton University Press, Princeton, N.J. 1967.
[39] A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
[40] M. P. Qian and D. Wang, On a system of hyperstable frequency locking persistence under white noise,Ergodic Theory Dynam,Syst. 20(2000), 547-555.
[41] M. Qian, W. Shen and J. Y. Zhang, Global Behavior in the dynamical equation of J-J type, J. Diff. Eq.77(1988), 315-333.
[42] M. Qian, W. Shen and J. Y. Zhang, Dynamical Behavior in the coupled systems of J-J type, J. Diff. Eq.88(1990), 175-212.
[43] M. Qian, S. Zhu and W. X. Qin, Dynamics in a chain of overdamped pendula driven by constant torques,SIAM J. Appl. Math. 57(1997), 294-305.
[44] M. Qian, W. X. Qin and S. Zhu, One dimensional global attractor for discretization of the damped drivensine-Gordon equation, Nonl. Anal., TMA. 34(1998), 941-951.
[45] M. Qian, S. F. Zhou and S. Zhu, One dimensional global attractor for the damped and driven sine-Gordonequation, Sci. China (Ser A) 41(2)(1998), 113.
[46] M. Qian, W. X. Qin, G. X. Wang and S. Zhu, Unbounded one dimensional global attractor for the dampedsine-Gordon equation, J. Nonlinear Sci. 10(2000), 417-432.
[47] W. Shen, Global attractor and rotation number of a class of nonlinear noisy oscillators, Discrete Contin.Dyn. S. 18(2007), 597-611.
[48] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, NewYork, 1998.
[49] G. Wang and S. Zhu, On Dimension of the global attractor for damped sine-Gordon equation, J. Math.Phy. 38 (6) (1997).
[50] J. Huang, The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with whitenoises, Physica D 233(2007) 83-94
[2] Birkhoff G D,Dynamical systems,AMS:Publications Providence.
[3]赵才地,非牛顿流与格点系统的渐近行为.上海大学,博士论文
[4] K.Ito On stochastic differential equations.Mem.Amer.Math.Soc.1951 4:51
[5] K.D.Elworthy,Stochastic dynamical systems and their ?ows.In A.Fried-man andM.Pinsky editors Stochastic Analysis Academic Press New York 1978 79 95.
[6] P.H.Baxendale Wiener processes on manifolds of maps.Proc.Roy.Soc.Edinburgh Sect.l980 87A:127 152.
[7] J.M.Bismut A generalized formula of Ito and some other properties of stochastic ?ows.Z.Wahrsch.Verve.Gebiete 1981 55:331 350.
[8] E.Pardoux and S.Peng Adapted solution of a backward stochastic differential equation.Systems ControlLett.1990 14:55 61.
[9] T.Caraballo et al.Upper semicontinuity of attractors for small random perbutations of dynamical syatems.Comm Partieal Differential Equations 1998 23:1557 1581.
[10] H.Crauel and F.Flandoli Attactors for random dynamical systems.Probab.Theory Relat.Fields 1994,100:365 393.
[11] Crauel H.etal.Random attractors.J.Differential Equations 1995 9:307 341.
[12] S.Zhou Attraetors for second order dissipative lattiee dynamical systems J.Differential Equa-tions 2002 179:605 624.
[13] P.W.Bares H.LiseiandK.Lu Attractors for stochastic lattice dynamical systems.Stochastic and Dynam-ics 2006 6:l 21
[14] PW.Bates K.Lu and B.Wang Attractors for lattice dynamical systems.Internationl Joumal of Bifurca-tionand Chaos 200l 11:143 153.
[15] C.D.Zhao and S.F.Zhou Sufficient conditions for the existence of global random attraetors for stochasticlattice dynamical systems and applieations.J.Math.Anal.Appl 2009 l:78 95.
[16] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998.
[17]张锦炎,常微分方程几何理论与分支问题,北京:北京大学出版社,1987.
[18] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on un-bounded domains, J. Differential Equations, 246(2009), 845-869.
[19] A. R. Bishop, K. Fesser, P. S. Lomdahl, and S. E. Trullinger, In?uence of solitons in the initial state onchaos in the driven damped sine-Gordon system, Phys. D 7 (1983), 259–279.
[20] A. R. Bishop and P. S. Lomdahl, Nonlinear dynamics in driven, damped sine-Gordon systems, Phys. D 18(1986), 54–66.
[21] A. N. Borodin, P. Salminen, Handbook of Brownian motion—facts and formulae. Second edition. Proba-bility and its Applications. Birkha¨user Verlag, Basel, 2002.
[22] T. Caraballo, P. E. Kloeden, and J. Real, Pullback and forward attractors for a damped wave equation withdelays, Stoch. Dyn. 4 (2004), 405–423.
[23] S.-N. Chow, W. Shen and H. M. Zhou, Dynamical order in systems of coupled noisy oscillators, J. Dyn.Diff. Eqns. 19(2007), 1007-1035.
[24] S.-N. Chow, X.-B. Lin and K. Lu, Smooth invariant foliations in infinite dimensional spaces, J. Diff. Eq.94(1991), 266-291.
[25] I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, 2002.
[26] H. Crauel, Random Probability Measure on Polish Spaces, Taylor & Francis:London, 2002.
[27] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, 1992.
[28] R. W. Dickey, Stability theory for the damped sine-Gordon equation, SIAM J. Appl. Math. 30 (1976),248–262.
[29] J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann.Probab. 31(2003), 2109 2135.
[30] X. M. Fan, Random attractor for a damped sine-Gordon equation with white noise, Pacific J. Math.216(2004), 63-76.
[31] X. M. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multi-plicative noise, Stoch. Anal. Appl. 24 (2006), 767–793.
[32] J. M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. PuresAppl. 66 (1987), 273-319.
[33] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI,1988.
[34] G. Kovacˇicˇand S. Wiggins, Orbits homoclinic to resonances, with an application to chaos in a model ofthe forced and damped sine-Gordon equation, Phys. D 57 (1992), 185–225
[35] M. Levi, F. C. Hoppensyeadt and W. L. Miranker, Dynamics of the Josephson junction, Quart. Appl.Math.7(1978), 167-198.
[36] R. Mansuy, M. Yor, Aspects of Brownian motion. Universitext. Springer-Verlag, Berlin, 2008.
[37] L. Nana, T. C. Kofan and E. Kaptouom, Subharmonic and homoclinic bifurcations in the driven anddamped sine-Gordon system, Phys. D 134 (1999), 61–74.
[38] E. Nelson Dynamical theories of Brownian motion. Princeton University Press, Princeton, N.J. 1967.
[39] A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
[40] M. P. Qian and D. Wang, On a system of hyperstable frequency locking persistence under white noise,Ergodic Theory Dynam,Syst. 20(2000), 547-555.
[41] M. Qian, W. Shen and J. Y. Zhang, Global Behavior in the dynamical equation of J-J type, J. Diff. Eq.77(1988), 315-333.
[42] M. Qian, W. Shen and J. Y. Zhang, Dynamical Behavior in the coupled systems of J-J type, J. Diff. Eq.88(1990), 175-212.
[43] M. Qian, S. Zhu and W. X. Qin, Dynamics in a chain of overdamped pendula driven by constant torques,SIAM J. Appl. Math. 57(1997), 294-305.
[44] M. Qian, W. X. Qin and S. Zhu, One dimensional global attractor for discretization of the damped drivensine-Gordon equation, Nonl. Anal., TMA. 34(1998), 941-951.
[45] M. Qian, S. F. Zhou and S. Zhu, One dimensional global attractor for the damped and driven sine-Gordonequation, Sci. China (Ser A) 41(2)(1998), 113.
[46] M. Qian, W. X. Qin, G. X. Wang and S. Zhu, Unbounded one dimensional global attractor for the dampedsine-Gordon equation, J. Nonlinear Sci. 10(2000), 417-432.
[47] W. Shen, Global attractor and rotation number of a class of nonlinear noisy oscillators, Discrete Contin.Dyn. S. 18(2007), 597-611.
[48] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, NewYork, 1998.
[49] G. Wang and S. Zhu, On Dimension of the global attractor for damped sine-Gordon equation, J. Math.Phy. 38 (6) (1997).
[50] J. Huang, The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with whitenoises, Physica D 233(2007) 83-94