# American Institute of Mathematical Sciences

May  2013, 18(3): 643-666. doi: 10.3934/dcdsb.2013.18.643

## Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing

 1 Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, United States

Received  May 2012 Revised  September 2012 Published  December 2012

This paper is concerned with the asymptotic behavior of solutions of the FitzHugh-Nagumo system on $\mathbb{R}^n$ driven by additive noise and deterministic non-autonomous forcing. We prove the system has a random attractor which pullback attracts all tempered random sets. We also prove the periodicity of the random attractor when the system is perturbed by time periodic forcing. The pullback asymptotic compactness of solutions is established by uniform estimates on the tails of solutions outside a large ball in $\mathbb{R}^n$.
Citation: Abiti Adili, Bixiang Wang. Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 643-666. doi: 10.3934/dcdsb.2013.18.643
##### References:
 [1] L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998).   Google Scholar [2] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Translated and revised from the 1989 Russian original by Babin, 25 (1989).   Google Scholar [3] J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations,, J. Nonl. Sci., 7 (1997), 475.  doi: 10.1007/s003329900037.  Google Scholar [4] P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems,, Stoch. Dyn., 6 (2006), 1.  doi: 10.1142/S0219493706001621.  Google Scholar [5] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains,, J. Differential Equations, 246 (2009), 845.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar [6] J. Bell, Some threshold results for models of myelinated nerves,, Mathematical Biosciences, 54 (1981), 181.  doi: 10.1016/0025-5564(81)90085-7.  Google Scholar [7] T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems,, Dynamics of Continuous, 10 (2003), 491.   Google Scholar [8] T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory,, Discrete Continuous Dynamical Systems B, 9 (2008), 525.  doi: 10.3934/dcdsb.2008.9.525.  Google Scholar [9] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness,, Discrete Contin. Dyn. Syst., 21 (2008), 415.  doi: 10.3934/dcds.2008.21.415.  Google Scholar [10] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar [11] T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion,, Nonlinear Analysis, 74 (2011), 3671.  doi: 10.1016/j.na.2011.02.047.  Google Scholar [12] I. Chueshow, "Monotone Random Systems - Theory and Applications,", Lecture Notes in Mathematics, 1779 (2002).  doi: 10.1007/b83277.  Google Scholar [13] I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems,, Dynamical Systems, 19 (2004), 127.  doi: 10.1080/1468936042000207792.  Google Scholar [14] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Th. Re. Fields, 100 (1994), 365.  doi: 10.1007/BF01193705.  Google Scholar [15] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Diff. Eqns., 9 (1997), 307.  doi: 10.1007/BF02219225.  Google Scholar [16] J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations,, Ann. Probab., 31 (2003), 2109.  doi: 10.1214/aop/1068646380.  Google Scholar [17] J. Duan, K. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations,, J. Dynam. Differential Equations, 16 (2004), 949.  doi: 10.1007/s10884-004-7830-z.  Google Scholar [18] J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary,, Comm. Math. Sci., 1 (2003), 133.   Google Scholar [19] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophys. J., 1 (1961), 445.   Google Scholar [20] F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise,, Stoch. Stoch. Rep., 59 (1996), 21.   Google Scholar [21] M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion,, J. Dynam. Differential Equations, 23 (2011), 671.  doi: 10.1007/s10884-011-9222-5.  Google Scholar [22] M. J. Garrido-Atienza, A. Ogrowsky and B. Schmalfuss, Random differential equations with random delays,, Stoch. Dyn., 11 (2011), 369.  doi: 10.1142/S0219493711003358.  Google Scholar [23] J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Mathematical Surveys and Monographs, 25 (1988).   Google Scholar [24] J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains,, Discrete and Continuous Dynamical Systems, 24 (2009), 855.  doi: 10.3934/dcds.2009.24.855.  Google Scholar [25] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors,, Proc. Royal Soc. London Serie A Math. Phys. Eng. Sci., 463 (2007), 163.  doi: 10.1098/rspa.2006.1753.  Google Scholar [26] Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for infinite-dimensional random dynamical systems in a Banach space,, Mem. Amer. Math. Soc., 206 (2010).  doi: 10.1090/S0065-9266-10-00574-0.  Google Scholar [27] Y. Lu and Z. Shao, Determining nodes for partly dissipative reaction diffusion systems,, Nonlinear Analysis, 54 (2003), 873.  doi: 10.1016/S0362-546X(03)00112-3.  Google Scholar [28] M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems,, SIAM J. Math. Anal., 20 (1989), 816.  doi: 10.1137/0520057.  Google Scholar [29] M. Marion, Inertial manifolds associated to partly dissipative reaction-diffusion systems,, J. Math. Anal. Appl., 143 (1989), 295.  doi: 10.1016/0022-247X(89)90043-7.  Google Scholar [30] S.-E. A. Mohammed, T. Zhang and H. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations,, Mem. Amer. Math. Soc., 196 (2008).   Google Scholar [31] J. Nagumo, S. Arimoto and S. Yosimzawa, An active pulse transmission line simulating nerve axon,, Proc. J. R. E., 50 (1964), 2061.   Google Scholar [32] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations,, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, (1992), 185.   Google Scholar [33] R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences, 143 (2002).   Google Scholar [34] Z. Shao, Existence of inertial manifolds for partly dissipative reaction diffusion systems in higher space dimensions,, J. Differential Equations, 144 (1998), 1.  doi: 10.1006/jdeq.1997.3383.  Google Scholar [35] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Second edition, 68 (1997).   Google Scholar [36] B. Wang, Attractors for reaction diffusion equations in unbounded domains,, Physica D, 128 (1999), 41.  doi: 10.1016/S0167-2789(98)00304-2.  Google Scholar [37] B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbbR^3$,, Transactions of American Mathematical Society, 363 (2011), 3639.  doi: 10.1090/S0002-9947-2011-05247-5.  Google Scholar [38] B. Wang, Random attractors for the Stochastic Benjamin-Bona-Mahony Equation on unbounded domains,, J. Differential Equations, 246 (2009), 2506.  doi: 10.1016/j.jde.2008.10.012.  Google Scholar [39] B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains,, Nonlinear Analysis, 71 (2009), 2811.  doi: 10.1016/j.na.2009.01.131.  Google Scholar [40] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems,, J. Differential Equations, 253 (2012), 1544.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar [41] B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems,, Electronic Journal of Differential Equations, 2009 (2009).   Google Scholar [42] B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains,, Electronic Journal of Differential Equations, 2012 (2012).   Google Scholar [43] B. Wang, Existence, stability and bifurcation of random periodic solutions of stochastic parabolic equations,, submitted for publication., ().   Google Scholar

show all references

##### References:
 [1] L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998).   Google Scholar [2] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Translated and revised from the 1989 Russian original by Babin, 25 (1989).   Google Scholar [3] J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations,, J. Nonl. Sci., 7 (1997), 475.  doi: 10.1007/s003329900037.  Google Scholar [4] P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems,, Stoch. Dyn., 6 (2006), 1.  doi: 10.1142/S0219493706001621.  Google Scholar [5] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains,, J. Differential Equations, 246 (2009), 845.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar [6] J. Bell, Some threshold results for models of myelinated nerves,, Mathematical Biosciences, 54 (1981), 181.  doi: 10.1016/0025-5564(81)90085-7.  Google Scholar [7] T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems,, Dynamics of Continuous, 10 (2003), 491.   Google Scholar [8] T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory,, Discrete Continuous Dynamical Systems B, 9 (2008), 525.  doi: 10.3934/dcdsb.2008.9.525.  Google Scholar [9] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness,, Discrete Contin. Dyn. Syst., 21 (2008), 415.  doi: 10.3934/dcds.2008.21.415.  Google Scholar [10] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar [11] T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion,, Nonlinear Analysis, 74 (2011), 3671.  doi: 10.1016/j.na.2011.02.047.  Google Scholar [12] I. Chueshow, "Monotone Random Systems - Theory and Applications,", Lecture Notes in Mathematics, 1779 (2002).  doi: 10.1007/b83277.  Google Scholar [13] I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems,, Dynamical Systems, 19 (2004), 127.  doi: 10.1080/1468936042000207792.  Google Scholar [14] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Th. Re. Fields, 100 (1994), 365.  doi: 10.1007/BF01193705.  Google Scholar [15] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Diff. Eqns., 9 (1997), 307.  doi: 10.1007/BF02219225.  Google Scholar [16] J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations,, Ann. Probab., 31 (2003), 2109.  doi: 10.1214/aop/1068646380.  Google Scholar [17] J. Duan, K. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations,, J. Dynam. Differential Equations, 16 (2004), 949.  doi: 10.1007/s10884-004-7830-z.  Google Scholar [18] J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary,, Comm. Math. Sci., 1 (2003), 133.   Google Scholar [19] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophys. J., 1 (1961), 445.   Google Scholar [20] F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise,, Stoch. Stoch. Rep., 59 (1996), 21.   Google Scholar [21] M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion,, J. Dynam. Differential Equations, 23 (2011), 671.  doi: 10.1007/s10884-011-9222-5.  Google Scholar [22] M. J. Garrido-Atienza, A. Ogrowsky and B. Schmalfuss, Random differential equations with random delays,, Stoch. Dyn., 11 (2011), 369.  doi: 10.1142/S0219493711003358.  Google Scholar [23] J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Mathematical Surveys and Monographs, 25 (1988).   Google Scholar [24] J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains,, Discrete and Continuous Dynamical Systems, 24 (2009), 855.  doi: 10.3934/dcds.2009.24.855.  Google Scholar [25] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors,, Proc. Royal Soc. London Serie A Math. Phys. Eng. Sci., 463 (2007), 163.  doi: 10.1098/rspa.2006.1753.  Google Scholar [26] Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for infinite-dimensional random dynamical systems in a Banach space,, Mem. Amer. Math. Soc., 206 (2010).  doi: 10.1090/S0065-9266-10-00574-0.  Google Scholar [27] Y. Lu and Z. Shao, Determining nodes for partly dissipative reaction diffusion systems,, Nonlinear Analysis, 54 (2003), 873.  doi: 10.1016/S0362-546X(03)00112-3.  Google Scholar [28] M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems,, SIAM J. Math. Anal., 20 (1989), 816.  doi: 10.1137/0520057.  Google Scholar [29] M. Marion, Inertial manifolds associated to partly dissipative reaction-diffusion systems,, J. Math. Anal. Appl., 143 (1989), 295.  doi: 10.1016/0022-247X(89)90043-7.  Google Scholar [30] S.-E. A. Mohammed, T. Zhang and H. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations,, Mem. Amer. Math. Soc., 196 (2008).   Google Scholar [31] J. Nagumo, S. Arimoto and S. Yosimzawa, An active pulse transmission line simulating nerve axon,, Proc. J. R. E., 50 (1964), 2061.   Google Scholar [32] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations,, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, (1992), 185.   Google Scholar [33] R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences, 143 (2002).   Google Scholar [34] Z. Shao, Existence of inertial manifolds for partly dissipative reaction diffusion systems in higher space dimensions,, J. Differential Equations, 144 (1998), 1.  doi: 10.1006/jdeq.1997.3383.  Google Scholar [35] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Second edition, 68 (1997).   Google Scholar [36] B. Wang, Attractors for reaction diffusion equations in unbounded domains,, Physica D, 128 (1999), 41.  doi: 10.1016/S0167-2789(98)00304-2.  Google Scholar [37] B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbbR^3$,, Transactions of American Mathematical Society, 363 (2011), 3639.  doi: 10.1090/S0002-9947-2011-05247-5.  Google Scholar [38] B. Wang, Random attractors for the Stochastic Benjamin-Bona-Mahony Equation on unbounded domains,, J. Differential Equations, 246 (2009), 2506.  doi: 10.1016/j.jde.2008.10.012.  Google Scholar [39] B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains,, Nonlinear Analysis, 71 (2009), 2811.  doi: 10.1016/j.na.2009.01.131.  Google Scholar [40] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems,, J. Differential Equations, 253 (2012), 1544.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar [41] B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems,, Electronic Journal of Differential Equations, 2009 (2009).   Google Scholar [42] B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains,, Electronic Journal of Differential Equations, 2012 (2012).   Google Scholar [43] B. Wang, Existence, stability and bifurcation of random periodic solutions of stochastic parabolic equations,, submitted for publication., ().   Google Scholar
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