June  2017, 22(4): 1587-1599. doi: 10.3934/dcdsb.2017077

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

Received  April 2016 Revised  November 2016 Published  February 2017

Fund Project: The author is supported by the National Natural Science Foundation of China grant 11371267,11571245 and the Basic Project of Sichuan Provincial Science and Technology Department grant 2016JY0204.

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.

Citation: Ji Shu. Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1587-1599. doi: 10.3934/dcdsb.2017077
References:
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L. Arnold, Random Dynamical Systems Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

P. BatesX. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.  doi: 10.1137/S0036141000374002.  Google Scholar

[3]

P. BatesH. Lisei and K. Lu, Attractors for stochastic lattice danymical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.  Google Scholar

[4]

P. BatesK. Lu and B. Wang, Attractors for lattice danymical systems, Int. J. Bifurcation Chaos, 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.  Google Scholar

[5]

P. BatesK. 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.  Google Scholar

[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.  Google Scholar

[7]

Z. BrzezniakM. Capinski and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Probab. Theory Relat. Fields, 95 (1993), 87-102.  doi: 10.1007/BF01197339.  Google Scholar

[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.  Google Scholar

[9]

S. Chow, Lattice dynamical systems, Lect. Notes Math., 1822 (2003), 1-102.  doi: 10.1007/978-3-540-45204-1_1.  Google Scholar

[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.  Google Scholar

[11]

S. ChowJ. 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.   Google Scholar

[12]

L. ChuaT. 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.  Google Scholar

[13]

I. Chueshov, Monotone Random Systems Theory and Applications Springer-Verlag, New York, 2002. doi: 10.1007/b83277.  Google Scholar

[14]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[15]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[16]

L. Decreusefond and A. Ustunel, Stochastic analysis of the fractional Brownian motion, Potential Anal., 10 (1999), 177-214.  doi: 10.1023/A:1008634027843.  Google Scholar

[17]

L. FabinyP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrodinger equations, Ⅰ, Bull. Sci. Engrg. Res. Lab. Waseda. Univ., 69 (1975), 51-62.   Google Scholar

[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.  Google Scholar

[22]

I. Fukuda and M. Tsutsumi, On Coupled Klein-Gordon-Schrodinger equations, Ⅲ, Math. Jpn., 24 (1979), 307-321.   Google Scholar

[23]

M. Garrido-AtienzaK. 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.  Google Scholar

[24]

M. Garrido-AtienzaPeter 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

X. HanW. 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.  Google Scholar

[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.  Google Scholar

[30]

R. Kapral, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.  doi: 10.1007/BF01192578.  Google Scholar

[31]

H. Kunita, Stochastic Flow and Stochastic Differential equations Cambridge University Press, Cambridge, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81.   Google Scholar

[37]

L. Pecora and T. Carrol, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.  doi: 10.1103/PhysRevLett.64.821.  Google Scholar

[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.   Google Scholar

[39]

Z. ShenS. 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.  Google Scholar

[40]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[41]

S. TindelC. 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.  Google Scholar

[42]

B. Wang, Dyanmics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003.  Google Scholar

[43]

X. WangS. 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.  Google Scholar

[44]

W. YanS. 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.  Google Scholar

[45]

M. Zahle, Integration with respect to fractal functions and stochastic calculus, Probab. Theory Relat. Fields, 111 (1998), 333-374.  doi: 10.1007/s004400050171.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

P. BatesX. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.  doi: 10.1137/S0036141000374002.  Google Scholar

[3]

P. BatesH. Lisei and K. Lu, Attractors for stochastic lattice danymical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.  Google Scholar

[4]

P. BatesK. Lu and B. Wang, Attractors for lattice danymical systems, Int. J. Bifurcation Chaos, 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.  Google Scholar

[5]

P. BatesK. 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.  Google Scholar

[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.  Google Scholar

[7]

Z. BrzezniakM. Capinski and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Probab. Theory Relat. Fields, 95 (1993), 87-102.  doi: 10.1007/BF01197339.  Google Scholar

[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.  Google Scholar

[9]

S. Chow, Lattice dynamical systems, Lect. Notes Math., 1822 (2003), 1-102.  doi: 10.1007/978-3-540-45204-1_1.  Google Scholar

[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.  Google Scholar

[11]

S. ChowJ. 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.   Google Scholar

[12]

L. ChuaT. 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.  Google Scholar

[13]

I. Chueshov, Monotone Random Systems Theory and Applications Springer-Verlag, New York, 2002. doi: 10.1007/b83277.  Google Scholar

[14]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[15]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[16]

L. Decreusefond and A. Ustunel, Stochastic analysis of the fractional Brownian motion, Potential Anal., 10 (1999), 177-214.  doi: 10.1023/A:1008634027843.  Google Scholar

[17]

L. FabinyP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrodinger equations, Ⅰ, Bull. Sci. Engrg. Res. Lab. Waseda. Univ., 69 (1975), 51-62.   Google Scholar

[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.  Google Scholar

[22]

I. Fukuda and M. Tsutsumi, On Coupled Klein-Gordon-Schrodinger equations, Ⅲ, Math. Jpn., 24 (1979), 307-321.   Google Scholar

[23]

M. Garrido-AtienzaK. 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.  Google Scholar

[24]

M. Garrido-AtienzaPeter 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

X. HanW. 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.  Google Scholar

[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.  Google Scholar

[30]

R. Kapral, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.  doi: 10.1007/BF01192578.  Google Scholar

[31]

H. Kunita, Stochastic Flow and Stochastic Differential equations Cambridge University Press, Cambridge, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81.   Google Scholar

[37]

L. Pecora and T. Carrol, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.  doi: 10.1103/PhysRevLett.64.821.  Google Scholar

[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.   Google Scholar

[39]

Z. ShenS. 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.  Google Scholar

[40]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[41]

S. TindelC. 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.  Google Scholar

[42]

B. Wang, Dyanmics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003.  Google Scholar

[43]

X. WangS. 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.  Google Scholar

[44]

W. YanS. 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.  Google Scholar

[45]

M. Zahle, Integration with respect to fractal functions and stochastic calculus, Probab. Theory Relat. Fields, 111 (1998), 333-374.  doi: 10.1007/s004400050171.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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