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Existence and multiplicity of traveling waves in a lattice dynamical system. (English) Zbl 0954.34029
Summary: The authors prove existence and multiplicity results of monotone traveling wave solutions to some lattice differential equations by using the monotone iteration method. The results include the model of cellular neural networks (CNN). In addition to the monotone traveling wave solutions, nonmonotone and oscillating traveling wave solutions in the delay type of CNN are obtained.

34C15Nonlinear oscillations, coupled oscillators (ODE)
92B20General theory of neural networks (mathematical biology)
34C12Monotone systems
34K11Oscillation theory of functional-differential equations
Full Text: DOI
[1] Afraimovich, V. S.; Nekorin, V. I.: Chaos of traveling waves in a discrete chain of diffusively coupled maps. Internat. J. Bifurcation chaos 4, 631-637 (1994) · Zbl 0870.58049
[2] Bell, J.: Some threshold results for models of myelinated nerves. Math. biosciences 54, 181-190 (1981) · Zbl 0454.92009
[3] Bell, J.; Cosner, C.: Threshold behavior and propagation for nonlinear differential--difference systems motivated by modeling myelinated axons. Quart. appl. Math. 42, 1-14 (1984) · Zbl 0536.34050
[4] Cahn, J. W.: Theory of crystal growth and interface motion in crystalline materials. Acta metallurgica 8, 554-562 (1960)
[5] Cahn, J. W.; Chow, S. -N.; Van Vleck, E. S.: Spatially discrete nonlinear diffusion equations. Rocky mountain J. Math. 25, 87-118 (1995) · Zbl 0833.65095
[6] Cahn, J. W.; Mallet-Paret, J.; Van Vleck, E. S.: Traveling wave solutions for systems of ODE’s on a two-dimensional spatial lattice. SIAM J. Appl. math. 59, 455-493 (1998) · Zbl 0917.34052
[7] Chow, S. -N.; Lin, X. B.; Mallet-Paret, J.: Transition layers for singular perturbed delay differential equations with monotone nonlinearities. J. dynam. Differential equations 1, 3-43 (1989) · Zbl 0684.34071
[8] Chow, S. -N.; Mallet-Paret, J.; Van Vleck, E. S.: Pattern formation and spatial chaos in spatially discrete evolution equations. Random comput. Dynamics 4, 109-178 (1996) · Zbl 0883.58020
[9] Chow, S. -N.; Mallet-Paret, J.; Shen, W.: Traveling waves in lattice dynamical systems. J. differential equations 149, 248-291 (1998) · Zbl 0911.34050
[10] Chow, S. -N.; Shen, W.: Stability and bifurcation of traveling wave solutions in coupled map lattices. J. dynamical systems appl. 4, 1-26 (1995) · Zbl 0821.34046
[11] Chua, L. O.; Roska, T.: The CNN paradigm. IEEE trans. Circuits and systems 40, 147-156 (1993) · Zbl 0800.92041
[12] Chua, L. O.; Yang, L.: Cellular neural networks: theory. IEEE trans. Circuits and systems 35, 1257-1272 (1988) · Zbl 0663.94022
[13] Chua, L. O.; Yang, L.: Cellular neural networks: applications. IEEE trans. Circuits and systems 35, 1273-1290 (1988)
[14] Erneux, T.; Nicolis, G.: Propagation waves in discrete bistable reaction--diffusion systems. Physica D 67, 237-244 (1993) · Zbl 0787.92010
[15] Fife, P.; Mcleod, J.: The approach of solutions of nonlinear diffusion equations to traveling front solutions. Arch. rational mech. Anal. 65, 333-361 (1977) · Zbl 0361.35035
[16] Gyori, I.; Ladas, G.: Oscillating theory of delay differential equations with applications. (1991)
[17] Hale, J. K.; Lunel, S. M. Verduyn: Introduction to functional differential equations. (1993) · Zbl 0787.34002
[18] Hankerson, D.; Zinner, B.: Wavefronts for cooperative tridigonal system of differential equations. J. dynam. Differential equations 5, 359-373 (1993) · Zbl 0777.34013
[19] Hsu, C. -H.; Lin, S. S.: Traveling waves in cellular neural networks. Int. J. Bifurcation and chaos 9, 1307-1319 (1999) · Zbl 0964.34033
[20] Hudson, H.; Zinner, B.: Existence of traveling waves for a generalized discrete Fisher’s equations. Comm. appl. Nonlinear anal. 1, 23-46 (1994) · Zbl 0859.34006
[21] Juang, J.; Lin, S. S.: Cellular neural networks: mosaic pattern and spatial chaos. SIAM J. Appl. math., 891-915 (2000) · Zbl 0947.34038
[22] J. Juang, and, S. S. Lin, Cellular neural networks: Defect pattern and spatial chaos, preprint. · Zbl 1193.34098
[23] Keener, J. P.: Propagation and its failure in coupled system of discrete excitable cells. SIAM J. Appl. math. 47, 556-572 (1987) · Zbl 0649.34019
[24] Laplante, J. P.; Erneux, T.: Propagation failure in arrays of coupled bistable chemical reactors. J. phys. Chem. 96, 4931-4934 (1992)
[25] J. Mallet-Paret, Stability and oscillation in nonlinear cyclic systems, in, Proceedings of Dynamical Systems Conference, Harvey Mudd College, California, June 1994, World Scientific, Singapore, to appear. · Zbl 0930.34024
[26] Mallet-Paret, J.: Spatial patterns, spatial chaos, and traveling waves in lattice differential equations. Stochastic and spatial structures of dynamical systems, 105-129 (1996) · Zbl 0980.37031
[27] Mallet-Paret, J.: The Fredholm alternative for functional differential equations of mixed type. J. dynam. Differential equations 11, 1-48 (1999) · Zbl 0927.34049
[28] Mallet-Paret, J.: The global structure of traveling waves in spatial discrete dynamical systems. J. dynam. Differential equations 11, 49-127 (1999) · Zbl 0921.34046
[29] Mallet-Paret, J.; Chow, S. -N.: Pattern formation and spatial chaos in lattice dynamical systems, II. IEEE trans. Circuits and systems 42, 752-756 (1995)
[30] Mallet-Paret, J.; Sell, G. R.: Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions. J. differential equations 125, 385-440 (1996) · Zbl 0849.34055
[31] Mallet-Paret, J.; Sell, G. R.: The Poincaré--Bendixson theorem for monotone cyclic feedback systems with delay. J. differential equations 125, 441-489 (1996) · Zbl 0849.34056
[32] Rudin, W.: Real and complex analysis. (1987) · Zbl 0925.00005
[33] Shen, W.: Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattices. SIAM J. Appl. math. 56, 1379-1399 (1996) · Zbl 0868.58059
[34] P. Thiran, K. R. Crounse, L. O. Chua, and M. Hasler, Pattern formation properties of autonomous cellular neural networks, IEEE Trans. Circuits and Systems421995, 757--774.
[35] Wu, J.; Zou, X.: Asymptotical and periodic boundary value problems of mixed fdes and wave solutions of lattice differential equations. J. differential equations 135, 315-357 (1997) · Zbl 0877.34046
[36] Zinner, B.: Stability of traveling wavefront solutions for the discrete Nagumo equation. SIAM J. Math. anal. 22, 1016-1020 (1991) · Zbl 0739.34060
[37] Zinner, B.: Existence of traveling wavefront solutions for discrete Nagumo equation. J. differential equations 96, 1-27 (1992) · Zbl 0752.34007
[38] Zinner, B.; Harris, G.; Hudson, W.: Traveling wavefronts for the discrete Fisher’s equation. J. differential equations 105, 46-62 (1993) · Zbl 0778.34006