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Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions
College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China |
In this paper, we study the asymptotic behavior of the stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions. With the properties of fractional Brownian motions, we prove the existence of a singleton sets random attractor.
References:
[1] |
L. Arnold, Random Dynamical Systems Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
P. Bates, X. Chen and A. Chmaj,
Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.
doi: 10.1137/S0036141000374002. |
[3] |
P. Bates, H. Lisei and K. Lu,
Attractors for stochastic lattice danymical systems, Stoch. Dyn., 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[4] |
P. Bates, K. Lu and B. Wang,
Attractors for lattice danymical systems, Int. J. Bifurcation Chaos, 11 (2001), 143-153.
doi: 10.1142/S0218127401002031. |
[5] |
P. Bates, K. Lu and B. Wang,
Random attractors for stochastic reaction-diffusion equations on unbounded domains, Journal of Differential Equations, 246 (2009), 845-869.
doi: 10.1016/j.jde.2008.05.017. |
[6] |
P. Biler,
Attractors for the system of Schrodinger and Klein-Gordon equations with Yukawa coupling, SIAM J. Math. Anal., 21 (1990), 1190-1212.
doi: 10.1137/0521065. |
[7] |
Z. Brzezniak, M. Capinski and F. Flandoli,
Pathwise global attractors for stationary random dynamical systems, Probab. Theory Relat. Fields, 95 (1993), 87-102.
doi: 10.1007/BF01197339. |
[8] |
T. Caraballo and K. Lu,
Attractors for stochastic lattice danymical systems with a multiplicativwe noise, Front. Math. China, 3 (2008), 317-335.
doi: 10.1007/s11464-008-0028-7. |
[9] |
S. Chow,
Lattice dynamical systems, Lect. Notes Math., 1822 (2003), 1-102.
doi: 10.1007/978-3-540-45204-1_1. |
[10] |
S. Chow and J. Paret,
Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Syst., 42 (1995), 746-751.
doi: 10.1109/81.473583. |
[11] |
S. Chow, J. Paret and E. Vleck,
Pattern formation and spatial chaos in lattice dynamical systems in spatially discrete evolution equations, Random Comput. Dyn., 4 (1996), 109-178.
|
[12] |
L. Chua, T. Roska and P. Venetianer,
The CNN paradigm is universal as the Turing machine, IEEE Trans. Circuits Syst., 40 (1993), 289-291.
doi: 10.1109/81.224308. |
[13] |
I. Chueshov,
Monotone Random Systems Theory and Applications Springer-Verlag, New York, 2002.
doi: 10.1007/b83277. |
[14] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dyn. Diff. Equ., 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[15] |
H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[16] |
L. Decreusefond and A. Ustunel,
Stochastic analysis of the fractional Brownian motion, Potential Anal., 10 (1999), 177-214.
doi: 10.1023/A:1008634027843. |
[17] |
L. Fabiny, P. Colet and R. Roy,
Coherence and phase dynamics of spatially coupled solid-state lasers, Phys. Rev. A, 47 (1993), 4287-4296.
doi: 10.1103/PhysRevA.47.4287. |
[18] |
X. Fan and Y. Wang,
Attractors for a second order nonautonomous lattice dynamical systems with nonlinear damping, Phys. Lett. A, 365 (2007), 17-27.
doi: 10.1016/j.physleta.2006.12.045. |
[19] |
F. Flandoli and B. Schmalfuss,
Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.
doi: 10.1080/17442509608834083. |
[20] |
I. Fukuda and M. Tsutsumi,
On coupled Klein-Gordon-Schrodinger equations, Ⅰ, Bull. Sci. Engrg. Res. Lab. Waseda. Univ., 69 (1975), 51-62.
|
[21] |
I. Fukuda and M. Tsutsumi,
On Coupled Klein-Gordon-Schrodinger equations, Ⅱ, J. Math. Anal. Appl., 66 (1978), 358-378.
doi: 10.1016/0022-247X(78)90239-1. |
[22] |
I. Fukuda and M. Tsutsumi,
On Coupled Klein-Gordon-Schrodinger equations, Ⅲ, Math. Jpn., 24 (1979), 307-321.
|
[23] |
M. Garrido-Atienza, K. Lu and B. Schmalfuss,
Random dynamical systems for stochastic equations driven by a fractional Brownian motion, Discrete Contin. Dyn. Syst. B, 14 (2010), 473-493.
doi: 10.3934/dcdsb.2010.14.473. |
[24] |
M. Garrido-Atienza, Peter E. Kloeden and A. Neuenkirch,
Discretization of stationary solutions of stochastic systems driven by a fractional Brownian motion, Appl. Math. Optim., 60 (2009), 151-172.
doi: 10.1007/s00245-008-9062-9. |
[25] |
M. Garrido-Atienza and B. Schmalfuss,
Ergodicity of the infinite dimensional fractional Brownian motion, J. Dyn. Differ. Equ., 23 (2011), 671-681.
doi: 10.1007/s10884-011-9222-5. |
[26] |
A. Gu and Y. Li,
Singleton sets random atractor for stochstic Fitzhugh-Nagumo lattice equations driven by fractional Brownian motions, Commu. Nonlin. Sci. Num. Simu., 19 (2014), 3928-3937.
doi: 10.1016/j.cnsns.2014.04.005. |
[27] |
B. Guo and Y. Li,
Attractors for Klein-Gordon-Schrodinger equations in $\mathbb{R}^{3}$, J. Differential Equations, 136 (1997), 356-377.
doi: 10.1006/jdeq.1996.3242. |
[28] |
X. Han, W. Shen and S. Zhou,
Random attractors for stochastic lattice dynamical system in weighted space, Journal of Differential Equations, 250 (2011), 1235-1266.
doi: 10.1016/j.jde.2010.10.018. |
[29] |
J. H. Huang,
The random attractor of stochstic Fitzhugh-Nagumo equations in infinite lattice with white noise, Physica D, 233 (2007), 83-94.
doi: 10.1016/j.physd.2007.06.008. |
[30] |
R. Kapral,
Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.
doi: 10.1007/BF01192578. |
[31] |
H. Kunita,
Stochastic Flow and Stochastic Differential equations Cambridge University Press, Cambridge, 1990. |
[32] |
K. Lu and B. Wang,
Global attractors for the Klein-Gordon-Schrodinger equations in unbounded domain, J. Differential Equations, 170 (2001), 281-316.
doi: 10.1006/jdeq.2000.3827. |
[33] |
Y. Lv and J. Sun,
Dynamical behavior for stochastic lattice systems, Chaos Solitons Fractals, 27 (2006), 1080-1090.
doi: 10.1016/j.chaos.2005.04.089. |
[34] |
Y. Lv and J. Sun,
Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations, Physica D, 221 (2006), 157-169.
doi: 10.1016/j.physd.2006.07.023. |
[35] |
B. Maslowski and B. Schmalfuss,
Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion, Stochastic Anal. Appl., 22 (2004), 1577-1607.
doi: 10.1081/SAP-200029498. |
[36] |
D. Nualart and A. Rascanu,
Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81.
|
[37] |
L. Pecora and T. Carrol,
Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.
doi: 10.1103/PhysRevLett.64.821. |
[38] |
B. Schmalfuss,
Backward cocycle and atttractors of stochastic differential equations, in International Semilar on Applied Mathematics-Nonlinear Dynamics:Attractor Approximation and Global Behavior (eds. V.Reitmann, T.Riedrich, and N.Koksch), Technishe Universität, Dresden, (1992), 185-192.
|
[39] |
Z. Shen, S. Zhou and W. Shen,
One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, Journal of Differential Equations, 248 (2010), 1432-1457.
doi: 10.1016/j.jde.2009.10.007. |
[40] |
R. Temam,
Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[41] |
S. Tindel, C. Tudor and F. Viens,
Stochastic evolution equations with fractional Brownian Motion, Probab. Theory Relat. Fields, 127 (2003), 186-204.
doi: 10.1007/s00440-003-0282-2. |
[42] |
B. Wang,
Dyanmics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.
doi: 10.1016/j.jde.2005.01.003. |
[43] |
X. Wang, S. Li and D. Xu,
Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494.
doi: 10.1016/j.na.2009.06.094. |
[44] |
W. Yan, S. Ji and Y. Li,
Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations, Phys. Lett. A, 373 (2009), 1268-1275.
doi: 10.1016/j.physleta.2009.02.019. |
[45] |
M. Zahle,
Integration with respect to fractal functions and stochastic calculus, Probab. Theory Relat. Fields, 111 (1998), 333-374.
doi: 10.1007/s004400050171. |
[46] |
C. Zhao and S. Zhou,
Compact kernel sections for nonautonomous Klein-Gordon-Schrodinger equations on infinite lattices, J. Math. Anal. Appl., 332 (2007), 32-56.
doi: 10.1016/j.jmaa.2006.10.002. |
[47] |
C. Zhao and S. Zhou,
Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95.
doi: 10.1016/j.jmaa.2008.12.036. |
[48] |
S. Zhou and W. Shi,
Attractors and dimension of dissipative lattice systems, J. Differential Equations, 224 (2006), 172-204.
doi: 10.1016/j.jde.2005.06.024. |
show all references
References:
[1] |
L. Arnold, Random Dynamical Systems Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
P. Bates, X. Chen and A. Chmaj,
Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.
doi: 10.1137/S0036141000374002. |
[3] |
P. Bates, H. Lisei and K. Lu,
Attractors for stochastic lattice danymical systems, Stoch. Dyn., 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[4] |
P. Bates, K. Lu and B. Wang,
Attractors for lattice danymical systems, Int. J. Bifurcation Chaos, 11 (2001), 143-153.
doi: 10.1142/S0218127401002031. |
[5] |
P. Bates, K. Lu and B. Wang,
Random attractors for stochastic reaction-diffusion equations on unbounded domains, Journal of Differential Equations, 246 (2009), 845-869.
doi: 10.1016/j.jde.2008.05.017. |
[6] |
P. Biler,
Attractors for the system of Schrodinger and Klein-Gordon equations with Yukawa coupling, SIAM J. Math. Anal., 21 (1990), 1190-1212.
doi: 10.1137/0521065. |
[7] |
Z. Brzezniak, M. Capinski and F. Flandoli,
Pathwise global attractors for stationary random dynamical systems, Probab. Theory Relat. Fields, 95 (1993), 87-102.
doi: 10.1007/BF01197339. |
[8] |
T. Caraballo and K. Lu,
Attractors for stochastic lattice danymical systems with a multiplicativwe noise, Front. Math. China, 3 (2008), 317-335.
doi: 10.1007/s11464-008-0028-7. |
[9] |
S. Chow,
Lattice dynamical systems, Lect. Notes Math., 1822 (2003), 1-102.
doi: 10.1007/978-3-540-45204-1_1. |
[10] |
S. Chow and J. Paret,
Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Syst., 42 (1995), 746-751.
doi: 10.1109/81.473583. |
[11] |
S. Chow, J. Paret and E. Vleck,
Pattern formation and spatial chaos in lattice dynamical systems in spatially discrete evolution equations, Random Comput. Dyn., 4 (1996), 109-178.
|
[12] |
L. Chua, T. Roska and P. Venetianer,
The CNN paradigm is universal as the Turing machine, IEEE Trans. Circuits Syst., 40 (1993), 289-291.
doi: 10.1109/81.224308. |
[13] |
I. Chueshov,
Monotone Random Systems Theory and Applications Springer-Verlag, New York, 2002.
doi: 10.1007/b83277. |
[14] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dyn. Diff. Equ., 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[15] |
H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[16] |
L. Decreusefond and A. Ustunel,
Stochastic analysis of the fractional Brownian motion, Potential Anal., 10 (1999), 177-214.
doi: 10.1023/A:1008634027843. |
[17] |
L. Fabiny, P. Colet and R. Roy,
Coherence and phase dynamics of spatially coupled solid-state lasers, Phys. Rev. A, 47 (1993), 4287-4296.
doi: 10.1103/PhysRevA.47.4287. |
[18] |
X. Fan and Y. Wang,
Attractors for a second order nonautonomous lattice dynamical systems with nonlinear damping, Phys. Lett. A, 365 (2007), 17-27.
doi: 10.1016/j.physleta.2006.12.045. |
[19] |
F. Flandoli and B. Schmalfuss,
Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.
doi: 10.1080/17442509608834083. |
[20] |
I. Fukuda and M. Tsutsumi,
On coupled Klein-Gordon-Schrodinger equations, Ⅰ, Bull. Sci. Engrg. Res. Lab. Waseda. Univ., 69 (1975), 51-62.
|
[21] |
I. Fukuda and M. Tsutsumi,
On Coupled Klein-Gordon-Schrodinger equations, Ⅱ, J. Math. Anal. Appl., 66 (1978), 358-378.
doi: 10.1016/0022-247X(78)90239-1. |
[22] |
I. Fukuda and M. Tsutsumi,
On Coupled Klein-Gordon-Schrodinger equations, Ⅲ, Math. Jpn., 24 (1979), 307-321.
|
[23] |
M. Garrido-Atienza, K. Lu and B. Schmalfuss,
Random dynamical systems for stochastic equations driven by a fractional Brownian motion, Discrete Contin. Dyn. Syst. B, 14 (2010), 473-493.
doi: 10.3934/dcdsb.2010.14.473. |
[24] |
M. Garrido-Atienza, Peter E. Kloeden and A. Neuenkirch,
Discretization of stationary solutions of stochastic systems driven by a fractional Brownian motion, Appl. Math. Optim., 60 (2009), 151-172.
doi: 10.1007/s00245-008-9062-9. |
[25] |
M. Garrido-Atienza and B. Schmalfuss,
Ergodicity of the infinite dimensional fractional Brownian motion, J. Dyn. Differ. Equ., 23 (2011), 671-681.
doi: 10.1007/s10884-011-9222-5. |
[26] |
A. Gu and Y. Li,
Singleton sets random atractor for stochstic Fitzhugh-Nagumo lattice equations driven by fractional Brownian motions, Commu. Nonlin. Sci. Num. Simu., 19 (2014), 3928-3937.
doi: 10.1016/j.cnsns.2014.04.005. |
[27] |
B. Guo and Y. Li,
Attractors for Klein-Gordon-Schrodinger equations in $\mathbb{R}^{3}$, J. Differential Equations, 136 (1997), 356-377.
doi: 10.1006/jdeq.1996.3242. |
[28] |
X. Han, W. Shen and S. Zhou,
Random attractors for stochastic lattice dynamical system in weighted space, Journal of Differential Equations, 250 (2011), 1235-1266.
doi: 10.1016/j.jde.2010.10.018. |
[29] |
J. H. Huang,
The random attractor of stochstic Fitzhugh-Nagumo equations in infinite lattice with white noise, Physica D, 233 (2007), 83-94.
doi: 10.1016/j.physd.2007.06.008. |
[30] |
R. Kapral,
Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.
doi: 10.1007/BF01192578. |
[31] |
H. Kunita,
Stochastic Flow and Stochastic Differential equations Cambridge University Press, Cambridge, 1990. |
[32] |
K. Lu and B. Wang,
Global attractors for the Klein-Gordon-Schrodinger equations in unbounded domain, J. Differential Equations, 170 (2001), 281-316.
doi: 10.1006/jdeq.2000.3827. |
[33] |
Y. Lv and J. Sun,
Dynamical behavior for stochastic lattice systems, Chaos Solitons Fractals, 27 (2006), 1080-1090.
doi: 10.1016/j.chaos.2005.04.089. |
[34] |
Y. Lv and J. Sun,
Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations, Physica D, 221 (2006), 157-169.
doi: 10.1016/j.physd.2006.07.023. |
[35] |
B. Maslowski and B. Schmalfuss,
Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion, Stochastic Anal. Appl., 22 (2004), 1577-1607.
doi: 10.1081/SAP-200029498. |
[36] |
D. Nualart and A. Rascanu,
Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81.
|
[37] |
L. Pecora and T. Carrol,
Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.
doi: 10.1103/PhysRevLett.64.821. |
[38] |
B. Schmalfuss,
Backward cocycle and atttractors of stochastic differential equations, in International Semilar on Applied Mathematics-Nonlinear Dynamics:Attractor Approximation and Global Behavior (eds. V.Reitmann, T.Riedrich, and N.Koksch), Technishe Universität, Dresden, (1992), 185-192.
|
[39] |
Z. Shen, S. Zhou and W. Shen,
One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, Journal of Differential Equations, 248 (2010), 1432-1457.
doi: 10.1016/j.jde.2009.10.007. |
[40] |
R. Temam,
Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[41] |
S. Tindel, C. Tudor and F. Viens,
Stochastic evolution equations with fractional Brownian Motion, Probab. Theory Relat. Fields, 127 (2003), 186-204.
doi: 10.1007/s00440-003-0282-2. |
[42] |
B. Wang,
Dyanmics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.
doi: 10.1016/j.jde.2005.01.003. |
[43] |
X. Wang, S. Li and D. Xu,
Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494.
doi: 10.1016/j.na.2009.06.094. |
[44] |
W. Yan, S. Ji and Y. Li,
Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations, Phys. Lett. A, 373 (2009), 1268-1275.
doi: 10.1016/j.physleta.2009.02.019. |
[45] |
M. Zahle,
Integration with respect to fractal functions and stochastic calculus, Probab. Theory Relat. Fields, 111 (1998), 333-374.
doi: 10.1007/s004400050171. |
[46] |
C. Zhao and S. Zhou,
Compact kernel sections for nonautonomous Klein-Gordon-Schrodinger equations on infinite lattices, J. Math. Anal. Appl., 332 (2007), 32-56.
doi: 10.1016/j.jmaa.2006.10.002. |
[47] |
C. Zhao and S. Zhou,
Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95.
doi: 10.1016/j.jmaa.2008.12.036. |
[48] |
S. Zhou and W. Shi,
Attractors and dimension of dissipative lattice systems, J. Differential Equations, 224 (2006), 172-204.
doi: 10.1016/j.jde.2005.06.024. |
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