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 and 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, Springer-Verlag, Berlin, 1998.

[2]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Translated and revised from the 1989 Russian original by Babin, Studies in Mathematics and its Applications, 25, North-Holland, Amsterdam, 1992.

[3]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonl. Sci., 7 (1997), 475-502. doi: 10.1007/s003329900037.

[4]

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

[5]

P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017.

[6]

J. Bell, Some threshold results for models of myelinated nerves, Mathematical Biosciences, 54 (1981), 181-190. doi: 10.1016/0025-5564(81)90085-7.

[7]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513.

[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), 525-539. doi: 10.3934/dcdsb.2008.9.525.

[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), 415-443 doi: 10.3934/dcds.2008.21.415.

[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), 439-455. doi: 10.3934/dcdsb.2010.14.439.

[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), 3671-3684. doi: 10.1016/j.na.2011.02.047.

[12]

I. Chueshow, "Monotone Random Systems - Theory and Applications," Lecture Notes in Mathematics, 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[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), 127-144. doi: 10.1080/1468936042000207792.

[14]

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

[15]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341. doi: 10.1007/BF02219225.

[16]

J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135. doi: 10.1214/aop/1068646380.

[17]

J. Duan, K. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972. doi: 10.1007/s10884-004-7830-z.

[18]

J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151.

[19]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.

[20]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stoch. Stoch. Rep., 59 (1996), 21-45.

[21]

M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681. doi: 10.1007/s10884-011-9222-5.

[22]

M. J. Garrido-Atienza, A. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388. doi: 10.1142/S0219493711003358.

[23]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.

[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), 855-882. doi: 10.3934/dcds.2009.24.855.

[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), 163-181. doi: 10.1098/rspa.2006.1753.

[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), vi+106 pp. doi: 10.1090/S0065-9266-10-00574-0.

[27]

Y. Lu and Z. Shao, Determining nodes for partly dissipative reaction diffusion systems, Nonlinear Analysis, 54 (2003), 873-884. doi: 10.1016/S0362-546X(03)00112-3.

[28]

M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal., 20 (1989), 816-844. doi: 10.1137/0520057.

[29]

M. Marion, Inertial manifolds associated to partly dissipative reaction-diffusion systems, J. Math. Anal. Appl., 143 (1989), 295-326. doi: 10.1016/0022-247X(89)90043-7.

[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), vi+105 pp.

[31]

J. Nagumo, S. Arimoto and S. Yosimzawa, An active pulse transmission line simulating nerve axon, Proc. J. R. E., 50 (1964), 2061-2070.

[32]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, (1992), 185-192.

[33]

R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.

[34]

Z. Shao, Existence of inertial manifolds for partly dissipative reaction diffusion systems in higher space dimensions, J. Differential Equations, 144 (1998), 1-43. doi: 10.1006/jdeq.1997.3383.

[35]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.

[36]

B. Wang, Attractors for reaction diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52. doi: 10.1016/S0167-2789(98)00304-2.

[37]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Transactions of American Mathematical Society, 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.

[38]

B. Wang, Random attractors for the Stochastic Benjamin-Bona-Mahony Equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537. doi: 10.1016/j.jde.2008.10.012.

[39]

B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Analysis, 71 (2009), 2811-2828. doi: 10.1016/j.na.2009.01.131.

[40]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.

[41]

B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electronic Journal of Differential Equations, 2009 (2009), 18 pp.

[42]

B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains, Electronic Journal of Differential Equations, 2012 (2012), 18 pp.

[43]

B. Wang, Existence, stability and bifurcation of random periodic solutions of stochastic parabolic equations, submitted for publication.

show all references

References:
[1]

L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.

[2]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Translated and revised from the 1989 Russian original by Babin, Studies in Mathematics and its Applications, 25, North-Holland, Amsterdam, 1992.

[3]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonl. Sci., 7 (1997), 475-502. doi: 10.1007/s003329900037.

[4]

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

[5]

P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017.

[6]

J. Bell, Some threshold results for models of myelinated nerves, Mathematical Biosciences, 54 (1981), 181-190. doi: 10.1016/0025-5564(81)90085-7.

[7]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513.

[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), 525-539. doi: 10.3934/dcdsb.2008.9.525.

[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), 415-443 doi: 10.3934/dcds.2008.21.415.

[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), 439-455. doi: 10.3934/dcdsb.2010.14.439.

[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), 3671-3684. doi: 10.1016/j.na.2011.02.047.

[12]

I. Chueshow, "Monotone Random Systems - Theory and Applications," Lecture Notes in Mathematics, 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[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), 127-144. doi: 10.1080/1468936042000207792.

[14]

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

[15]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341. doi: 10.1007/BF02219225.

[16]

J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135. doi: 10.1214/aop/1068646380.

[17]

J. Duan, K. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972. doi: 10.1007/s10884-004-7830-z.

[18]

J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151.

[19]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.

[20]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stoch. Stoch. Rep., 59 (1996), 21-45.

[21]

M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681. doi: 10.1007/s10884-011-9222-5.

[22]

M. J. Garrido-Atienza, A. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388. doi: 10.1142/S0219493711003358.

[23]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.

[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), 855-882. doi: 10.3934/dcds.2009.24.855.

[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), 163-181. doi: 10.1098/rspa.2006.1753.

[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), vi+106 pp. doi: 10.1090/S0065-9266-10-00574-0.

[27]

Y. Lu and Z. Shao, Determining nodes for partly dissipative reaction diffusion systems, Nonlinear Analysis, 54 (2003), 873-884. doi: 10.1016/S0362-546X(03)00112-3.

[28]

M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal., 20 (1989), 816-844. doi: 10.1137/0520057.

[29]

M. Marion, Inertial manifolds associated to partly dissipative reaction-diffusion systems, J. Math. Anal. Appl., 143 (1989), 295-326. doi: 10.1016/0022-247X(89)90043-7.

[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), vi+105 pp.

[31]

J. Nagumo, S. Arimoto and S. Yosimzawa, An active pulse transmission line simulating nerve axon, Proc. J. R. E., 50 (1964), 2061-2070.

[32]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, (1992), 185-192.

[33]

R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.

[34]

Z. Shao, Existence of inertial manifolds for partly dissipative reaction diffusion systems in higher space dimensions, J. Differential Equations, 144 (1998), 1-43. doi: 10.1006/jdeq.1997.3383.

[35]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.

[36]

B. Wang, Attractors for reaction diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52. doi: 10.1016/S0167-2789(98)00304-2.

[37]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Transactions of American Mathematical Society, 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.

[38]

B. Wang, Random attractors for the Stochastic Benjamin-Bona-Mahony Equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537. doi: 10.1016/j.jde.2008.10.012.

[39]

B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Analysis, 71 (2009), 2811-2828. doi: 10.1016/j.na.2009.01.131.

[40]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.

[41]

B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electronic Journal of Differential Equations, 2009 (2009), 18 pp.

[42]

B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains, Electronic Journal of Differential Equations, 2012 (2012), 18 pp.

[43]

B. Wang, Existence, stability and bifurcation of random periodic solutions of stochastic parabolic equations, submitted for publication.

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