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 and Continuous Dynamical Systems - B, 2017, 22 (4) : 1587-1599. doi: 10.3934/dcdsb.2017077
References:
[1]

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

[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.

[3]

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

[4]

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

[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.

[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. 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.

[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. 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. 

[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.

[13]

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

[14]

H. CrauelA. 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. 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.

[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-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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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. 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.

[3]

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

[4]

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

[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.

[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. 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.

[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. 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. 

[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.

[13]

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

[14]

H. CrauelA. 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. 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.

[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-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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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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