Attractors for stochastic lattice dynamical systems with a multiplicative noise

Abstract

In this paper, we consider a stochastic lattice differential equation with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and multiplicative white noise at each node. We prove the existence of a compact global random attractor which, pulled back, attracts tempered random bounded sets.

Keywords

Stochastic lattice differential equation / random attractor / multiplicative noise

References

[1.] Afraimovich V. S., Nekorkin V. I. Chaos of traveling waves in a discrete chain of diffusively coupled maps. Int J Bifur Chaos, 1994, 4: 631-637.

[2.] Arnold L. Random Dynamical Systems, 1998, Berlin: Springer-Verlag.

[3.] Bates P. W., Chmaj A. A discrete convolution model for phase transitions. Arch Ration Mech Anal, 1999, 150(4): 281-305.

[4.] Bates P. W., Lisei H., Lu K. Attractors for stochastic lattice dynamical systems. Stochastics & Dynamics, 2006, 6(1): 1-21.

[5.] Bates P. W., Lu K., Wang B. Attractors for lattice dynamical systems. Int J Bifur Chaos, 2001, 11: 143-153.

[6.] Bell J. Some threshhold results for models of myelinated nerves. Mathematical Biosciences, 1981, 54: 181-190.

[7.] Bell J., Cosner C. Threshold behaviour and propagation for nonlinear differentialdifference systems motivated by modeling myelinated axons. Quarterly Appl Math, 1984, 42: 1-14.

[8.] Caraballo T., Kloeden P. E., Schmalfuß B. Exponentially stable stationary solutions for stochastic evolution equations and their perturbation. Applied Mathematics and Optimization, 2004, 50: 183-207.

[9.] Caraballo T., Lukaszewicz G., Real J. Pullback attractors for asymptotically compact nonautonomous dynamical systems. Nonlinear Analysis TMA, 2006, 64(3): 484-498.

[10.] Chow S.-N., Mallet-Paret J. Pattern formulation and spatial chaos in lattice dynamical systems: I. IEEE Trans Circuits Syst, 1995, 42: 746-751.

[11.] Chow S.-N., Mallet-Paret J., Shen W. Traveling waves in lattice dynamical systems. J Diff Eq, 1998, 149: 248-291.

[12.] Chow S.-N., Mallet-Paret J., Van Vleck E. S. Pattern formation and spatial chaos in spatially discrete evolution equations. Random Computational Dynamics, 1996, 4: 109-178.

[13.] Chow S.-N., Shen W. Dynamics in a discrete Nagumo equation: Spatial topological chaos. SIAM J Appl Math, 1995, 55: 1764-1781.

[14.] Chua L. O., Roska T. The CNN paradigm. IEEE Trans Circuits Syst, 1993, 40: 147-156.

[15.] Chua L. O., Yang L. Cellular neural networks: Theory. IEEE Trans Circuits Syst, 1988, 35: 1257-1272.

[16.] Chua L. O., Yang L. Cellular neural networks: Applications. IEEE Trans Circuits Syst, 1988, 35: 1273-1290.

[17.] Crauel H. Random point attractors versus random set attractors. J London Math Soc, 2002, 63: 413-427.

[18.] Crauel H., Debussche A., Flandoli F. Random Attractors. J Dyn Diff Eq, 1997, 9: 307-341.

[19.] Crauel H., Flandoli F. Attractors for random dynamical systems. Probab Theory Relat Fields, 1994, 100: 365-393.

[20.] Dogaru R., Chua L. O. Edge of chaos and local activity domain of Fitz-Hugh-Nagumo equation. Int J Bifurcation and Chaos, 1988, 8: 211-257.

[21.] Erneux T., Nicolis G. Propagating waves in discrete bistable reaction diffusion systems. Physica D, 1993, 67: 237-244.

[22.] Flandoli F., Lisei H. Stationary conjugation of flows for parabolic SPDEs with multiplicative noise and some applications. Stoch Anal Appl, 2004, 22: 1385-1420.

[23.] Flandoli F., Schmalfuß B. Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise. Stochastics and Stochastic Rep, 1996, 59: 21-45.

[24.] Imkeller P., Schmalfuß B. The conjugacy of stochastic and random differential equations and the existence of global attractors. J Dyn Diff Eq, 2001, 13: 215-249.

[25.] Kapval R. Discrete models for chemically reacting systems. J Math Chem, 1991, 6: 113-163.

[26.] Keener J. P. Propagation and its failure in coupled systems of discrete excitable cells. SIAM J Appl Math, 1987, 47: 556-572.

[27.] Keener J. P. The effects of discrete gap junction coupling on propagation in myocardium. J Theor Biol, 1991, 148: 49-82.

[28.] Laplante J. P., Erneux T. Propagating failure in arrays of coupled bistable chemical reactors. J Phys Chem, 1992, 96: 4931-4934.

[29.] Mallet-Paret J. The global structure of traveling waves in spatially discrete dynamical systems. J Dynam Differential Equations, 1999, 11(1): 49-127.

[30.] Pérez-Muñuzuri A., Pérez-Muñuzuri V., Pérez-Villar V. Spiral waves on a 2-d array of nonlinear circuits. IEEE Trans Circuits Syst, 1993, 40: 872-877.

[31.] Rashevsky N. Mathematical Biophysics, 1960, New York: Dover Publications, Inc.

[32.] Ruelle D. Characteristic exponents for a viscous fluid subjected to time dependent forces. Commu Math Phys, 1984, 93: 285-300.

[33.] Scheutzow M. Comparison of various concepts of a random attractor: A case study. Arch Math, 2002, 78: 233-240.

[34.] Scott A. C. Analysis of a myelinated nerve model. Bull Math Biophys, 1964, 26: 247-254.

[35.] Shen W. Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattices. SIAM J Appl Math, 1996, 56: 1379-1399.

[36.] Zinner B. Existence of traveling wavefront solutions for the discrete Nagumo equation. J Diff Eq, 1992, 96: 1-27.