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Traveling waves in lattice dynamical systems. (English) Zbl 0911.34050

Summary: The authors study the existence and stability of travelling waves in lattice dynamical systems, in particular, in lattice ordinary differential equations (lattice ODEs) and in coupled map lattices (CMLs). Instead of employing the moving coordinate approach as for partial differential equations they construct a local coordinate system around a traveling wave solution to a lattice ODE, analogous to the local coordinate system around a periodic solution to an ODE. In this coordinate system the lattice ODE becomes a nonautonomous periodic differential equation, and the traveling wave corresponds to a periodic solution to this equation. The authors prove the asymptotic stability with asymptotic phase shift of the traveling wave solution under appropriate spectral conditions. It is shown the existence of traveling waves in CMLs which arise as time-discretizations of lattice ODEs. Finally, these results are applied to the discrete Nagumo equation. \(\copyright\) 1998 Academic Press.

MSC:

34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
37-XX Dynamical systems and ergodic theory
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34D10 Perturbations of ordinary differential equations
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[1] Afraimovich, V. S.; Nekorkin, V. I., Chaos of traveling waves in a discrete chain of diffusively coupled maps, Int. J. Bifur. Chaos Appl. Sci. Engrg., 4, 631-637 (1994) · Zbl 0870.58049
[2] Afraimovich, V. S.; Pesin, Ya. G., Traveling waves in lattice models of multi-dimensional and multi-component media, I. General hyperbolic properties, Nonlinearity, 6, 429-455 (1993) · Zbl 0773.76006
[3] Afraimovich, V. S.; Pesin, Ya. G., Traveling waves in lattice models of multi-dimensional and multi-component media, II. Ergodic properties and dimension, Chaos, 3, 233-241 (1993) · Zbl 1055.37584
[4] Alexander, J. C.; Gardner, R.; Jones, C., A topological invariant arising in the stability analysis of traveling waves, J. Reine Angew. Math., 410, 167-212 (1990) · Zbl 0705.35070
[5] N. D. Alikakos, P. W. Bates, X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc.; N. D. Alikakos, P. W. Bates, X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc. · Zbl 0929.35067
[6] N. D. Alikakos, P. W. Bates, X. Chen, Traveling waves in a time periodic structure and a singular perturbation problem; N. D. Alikakos, P. W. Bates, X. Chen, Traveling waves in a time periodic structure and a singular perturbation problem
[7] Aronson, D. G.; Golubitsky, M.; Mallet-Paret, J., Ponies on a merry-go-round in large arrays of Josephson junctions, Nonlinearity, 4, 903-910 (1991) · Zbl 0732.58031
[8] D. G. Aronson, Y. S. Huang, Limit and uniqueness of discrete rotating waves in large arrays Josephson junctions, Nonlinearity, 7, 1994, 777, 804; D. G. Aronson, Y. S. Huang, Limit and uniqueness of discrete rotating waves in large arrays Josephson junctions, Nonlinearity, 7, 1994, 777, 804 · Zbl 0803.34026
[9] Bates, P. W.; Fife, P. C.; Ren, X.; Wang, X., Traveling waves in a convolution model for phase transitions, Arch. Rat. Mech. Anal., 138, 105-136 (1997) · Zbl 0889.45012
[10] P. W. Bates, K. Lu, C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc.; P. W. Bates, K. Lu, C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc. · Zbl 1023.37013
[11] P. W. Bates, K. Lu, C. Zeng, Invariant foliations near normally hyperbolic invariant manifolds for semiflows, Trans. Amer. Math. Soc.; P. W. Bates, K. Lu, C. Zeng, Invariant foliations near normally hyperbolic invariant manifolds for semiflows, Trans. Amer. Math. Soc. · Zbl 0964.37018
[12] Bayliss, A.; Matkowsky, B. J., Fronts, relaxation oscillations, and periodic doubling in solid fuel combustion, J. Comput. Phys., 71, 147-168 (1987) · Zbl 0616.65133
[13] Bell, J., Some threshold results for models of myelinated nerves, Math. Biosci., 54, 181-190 (1981) · Zbl 0454.92009
[14] 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
[15] Benzoni-Gavage, S., Semi-discrete shock profiles for hyperbolic systems of conservation laws, Phys. D, 115, 109-123 (1998) · Zbl 0934.65098
[16] Berestycki, H.; Nicolaenko, B.; Scheurer, B., Traveling wave solutions to combustion models and their singular limits, SIAM J. Math. Anal., 16, 1207-1242 (1985) · Zbl 0596.76096
[17] Cahn, J. W., Theory of crystal growth and interface motion in crystalline materials, Acta Metallurgica, 8, 554-562 (1960)
[18] J. W. Cahn, J. Mallet-Paret, E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math.; J. W. Cahn, J. Mallet-Paret, E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math. · Zbl 0917.34052
[19] Chen, X., Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2, 125-160 (1997) · Zbl 1023.35513
[20] Chow, S.-N.; Lin, X.-B.; Lu, K., Smooth invariant foliations in infinite dimensional spaces, J. Differential Equations, 94, 266-291 (1991) · Zbl 0749.58043
[21] Chow, S.-N.; Mallet-Paret, J., Pattern formation and spatial chaos in lattice dynamical systems: I, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42, 746-751 (1995)
[22] Chow, S.-N.; Mallet-Paret, J.; Van Vleck, E. S., Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comput. Dynam., 4, 109-178 (1996) · Zbl 0883.58020
[23] Chow, S.-N.; Mallet-Paret, J.; Van Vleck, E. S., Dynamics of lattice differential equations, Int. J. Bifur. Chaos Appl. Sci. Engrg., 6, 1605-1621 (1996) · Zbl 1005.39504
[24] Chow, S.-N.; Shen, W., Dynamics in a discrete Nagumo equation: spatial topological chaos, SIAM J. Appl. Math., 55, 1764-1781 (1995) · Zbl 0840.34012
[25] Chow, S.-N.; Shen, W., Stability and bifurcation of traveling wave solutions in coupled map lattices, Dynam. Systems Appl., 4, 1-25 (1995) · Zbl 0821.34046
[26] Chow, S.-N.; Shen, W., A free boundary problem related to condensed two-phase combustion. Part I. Semigroup, J. Differential Equations, 108, 342-389 (1994) · Zbl 0812.35162
[27] Chow, S.-N.; Shen, W., A free boundary problem related to condensed two-phase combustion. Part II. Stability and bifurcation, J. Differential Equations, 108, 390-423 (1994) · Zbl 0812.35163
[28] Chua, L. O.; Roska, T., The CNN paradigm, IEEE Trans. Circuits Systems, 40, 147-156 (1993) · Zbl 0800.92041
[29] Chua, L. O.; Yang, L., Cellular neural networks: Theory, IEEE Trans. Circuits Systems, 35, 1257-1272 (1988) · Zbl 0663.94022
[30] Chua, L. O.; Yang, L., Cellular neural networks: Applications, IEEE Trans. Circuits Systems, 35, 1273-1290 (1988)
[31] Conley, C. C.; Gardner, R., An application of generalized Morse index to traveling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33, 319-343 (1984) · Zbl 0565.58016
[32] Cook, H. E.; de Fontaine, D.; Hilliard, J. E., A model for diffusion on cubic lattices and its application to the early stages of ordering, Acta Metallurgica, 17, 765-773 (1969)
[33] De Masi, A.; Gobron, T.; Presutti, E., Traveling fronts in non-local evolution equations, Arch. Rational Mech. Anal., 132, 143-205 (1995) · Zbl 0847.45008
[34] Edelstein, I.; Mitjagin, B.; Semenov, E., The linear groups of \(CL_1\), Bull. Acad. Polonaise Sci., 18, 27-33 (1970) · Zbl 0192.47401
[35] Elmer, C. E.; Van Vleck, E. S., Computation of traveling waves for spatially discrete bistable reaction-diffusion equations, Appl. Numer. Math., 20, 157-169 (1996) · Zbl 0856.65116
[36] Ermentrout, G. B., Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators, SIAM J. Appl. Math., 52, 1665-1687 (1992) · Zbl 0786.45005
[37] Ermentrout, G. B.; Kopell, N., Inhibition-produced patterning in chains of coupled nonlinear oscillators, SIAM J. Appl. Math., 54, 478-507 (1994) · Zbl 0811.92004
[38] Ermentrout, G. B.; McLeod, J. B., Existence and uniqueness of traveling waves for a neural network, Proc. Roy. Soc. Edinburg Sect. A, 123A, 461-478 (1993) · Zbl 0797.35072
[39] Erneux, T.; Nicolis, G., Propagating waves in discrete bistable reaction diffusion systems, Phys. D, 67, 237-244 (1993) · Zbl 0787.92010
[40] Fenichel, N., Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21, 193-226 (1971) · Zbl 0246.58015
[41] Fife, P.; McLeod, J. B., The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rational Mech. Anal., 65, 333-361 (1977) · Zbl 0361.35035
[42] Fife, P. C.; Wang, X., A convolution model for interfacial motion: the generation and propagation of internal layers in higher space dimensions, Adv. Differential Equations, 3, 85-110 (1998) · Zbl 0954.35087
[43] Firth, W. J., Optical memory and spatial chaos, Phys. Rev. Lett., 61, 329-332 (1988)
[44] Gardner, R., Existence and stability of traveling wave solutions of competition models: a degree theoretic approach, J. Differential Equations, 44, 343-364 (1982) · Zbl 0446.35012
[45] Hale, J. K., Ordinary Differential Equations (1980), Krieger: Krieger Melbourne · Zbl 0186.40901
[46] Hankerson, D.; Zinner, B., Wavefronts for a cooperative tridiagonal system of differential equations, J. Dyn. Differential Equations, 5, 359-373 (1993) · Zbl 0777.34013
[47] Henry, D., Geometric Theory of Semilinear Parabolic Equations. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., 840 (1981), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0456.35001
[48] Hillert, M., A solid-solution model for inhomogeneous systems, Acta Metallurgica, 9, 525-535 (1961)
[49] Kapral, R., Discrete models for chemically reacting systems, J. Math. Chem., 6, 113-163 (1991)
[50] Keener, J. P., Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47, 556-572 (1987) · Zbl 0649.34019
[51] Keener, J. P., The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol., 148, 49-82 (1991)
[52] Kopell, N.; Ermentrout, G. B.; Williams, T. L., On chains of oscillators forced at one end, SIAM J. Appl. Math., 51, 1397-1417 (1991) · Zbl 0748.34020
[53] Kopell, N.; Zhang, W.; Ermentrout, G. B., Multiple coupling in chains of oscillators, SIAM J. Math. Anal., 21, 935-953 (1990) · Zbl 0703.34055
[54] Kuiper, M. H., The homotopy type of the unitary group of Hilbert space, Topology, 3, 19-40 (1965) · Zbl 0129.38901
[55] Laplante, J. P.; Erneux, T., Propagation failure in arrays of coupled bistable chemical reactors, J. Phys. Chem., 96, 4931-4934 (1992)
[56] T.-P. Liu, S.-H. Yu, Continuum shock profiles for discrete conservation laws, I. Construction, Comm. Pure. Appl. Math.; T.-P. Liu, S.-H. Yu, Continuum shock profiles for discrete conservation laws, I. Construction, Comm. Pure. Appl. Math. · Zbl 0933.35137
[57] Lorenz, J., Numerics of invariant manifolds and attractors, Chaotic Numerics (Geelong, 1993) (1994), p. 185-202 · Zbl 0811.65051
[58] Mallet-Paret, J., Stability and oscillation in nonlinear cyclic systems, (Martelli, M.; Cooke, K.; Cumberbatch, E.; Tang, B.; Thieme, H., Differential Equations and Applications to Biology and Industry (1996), World Scientific: World Scientific Singapore), 337-346 · Zbl 0930.34024
[59] Mallet-Paret, J., Spatial patterns, spatial chaos, and traveling waves in lattice differ- ential equations, (van Strien, S. J.; Verduyn Lunel, S. M., Stochastic and Spatial Structures of Dynamical Systems (1996), North-Holland: North-Holland Amsterdam), 105-129 · Zbl 0980.37031
[60] J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dyn. Differential Equations; J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dyn. Differential Equations · Zbl 0927.34049
[61] J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dyn. Differential Equations; J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dyn. Differential Equations · Zbl 0921.34046
[62] Mallet-Paret, J.; Chow, S.-N., Pattern formation and spatial chaos in lattice dynamical systems: II, IEEE Trans. Circuits Systems, 42, 752-756 (1995)
[63] Neubauer, G., Der Homotopietyp der Automorphismengruppe in den Räumen\(l_p\)und \(c_0\), Math. Ann., 174, 33-40 (1967) · Zbl 0158.14402
[64] Nishiura, Y., Singular limit approach to stability and bifurcation for bistable reaction diffusion systems, Rocky Mountain J. Math., 21, 727-767 (1991) · Zbl 0804.35059
[65] Pérez-Muñuzuri, A.; Pérez-Muñuzuri, V.; Pérez-Villar, V.; Chua, L. O., Spiral waves on a 2-d array of nonlinear circuits, IEEE Trans. Circuits Systems, 40, 872-877 (1993) · Zbl 0844.93056
[66] Pérez-Muñuzuri, V.; Pérez-Villar, V.; Chua, L. O., Propagation failure in linear arrays of Chua’s circuits, Int. J. Bifur. Chaos, 2, 403-406 (1992) · Zbl 0875.94136
[67] Sattinger, D. H., On the stability of waves of nonlinear parabolic systems, Adv. Math., 22, 312-355 (1976) · Zbl 0344.35051
[68] Shen, W., Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattices, SIAM J. Appl. Math., 56, 1379-1399 (1996) · Zbl 0868.58059
[69] Terman, D., Traveling wave solutions arising from a two-step combustion model, SIAM J. Math. Anal., 19, 1057-1080 (1988) · Zbl 0691.35051
[70] Volpert, A. I.; Volpert, V. A.; Volpert, V. A., Traveling Wave Solutions of Parabolic Systems. Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, 140 (1994), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0835.35048
[71] Weinberger, H. F., Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13, 353-396 (1982) · Zbl 0529.92010
[72] Winslow, R. L.; Kimball, A. L.; Varghese, A., Simulating cardiac sinus and atrial network dynamics on the Connection Machine, Phys. D, 64, 281-298 (1993) · Zbl 0769.92010
[73] Xin, X., Existence and uniqueness of travelling waves in a reaction-diffusion equation with combustion nonlinearity, Indiana Univ. Math. J., 40, 985-1008 (1991) · Zbl 0727.35070
[74] S.-H. Yu, Existence of discrete shock profiles for the Lax-Wendroff scheme, Stanford, 1994; S.-H. Yu, Existence of discrete shock profiles for the Lax-Wendroff scheme, Stanford, 1994
[75] Zinner, B., Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22, 1016-1020 (1991) · Zbl 0739.34060
[76] Zinner, B., Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Differential Equations, 96, 1-27 (1992) · Zbl 0752.34007
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