Random attractors for  non-autonomousstochastic  FitzHugh-Nagumo systems with multiplicative noise

Abiti Adili; Bixiang Wang

doi:10.3934/proc.2013.2013.1

Article Contents

2013, Volume 2013: 1-10. Doi: 10.3934/proc.2013.2013.1 open access

This issue Previous Article Next Article Asymptotic behavior of a ratio-dependent predator-prey system with disease in the prey

Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise

Abiti Adili^1, and
Bixiang Wang^1,

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

Received: August 31, 2012

Revised: November 30, 2012

Published: October 2013

Abstract Related Papers Cited by

Abstract

In this paper, we prove the existence and uniqueness of random attractors for the FitzHugh-Nagumo system defined on $\mathbb{R}^n$ driven by both deterministic non-autonomous forcing and multiplicative noise. The periodicity of random attractors is established when the system is perturbed by time periodic forcing. We also prove the upper semicontinuity of random attractors when the intensity of noise approaches zero.

Keywords:

Mathematics Subject Classification: Primary: 37L55; Secondary: 37L30, 35R60, 60H15.

Citation:

Related Papers

Cited by

References

[1]	P.W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.
[2]	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.
[3]	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.
[4]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393.
[5]	J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151.
[6]	F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.
[7]	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.
[8]	B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012),1544-1583.
[9]	B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, arXiv:1205.4658v1, 2012.
[10]	B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete and Continuous Dynamical Systems, Series A, 34 (2014), 269-300.