Traveling Waves in Lattice Dynamical Systems

Journal of Differential Equations

Volumn 149, Issue 2, 1998, Pages 248-291

  Chow, Shui Nee ^a,b   Mallet Paret, John ^c   Shen, Wenxian ^d  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^b NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^c BROWN UNIVERSITY   (United States)

^d AUBURN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001379558     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1998.3478     Document Type: Article

References (76)

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. 4:1994;631-637.
2. Afraimovich V. S., Pesin Ya. G. Traveling waves in lattice models of multi-dimensional and multi-component media, I. General hyperbolic properties. Nonlinearity. 6:1993;429-455.
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. 3:1993;233-241.
4. Alexander J. C., Gardner R., Jones C. A topological invariant arising in the stability analysis of traveling waves. J. Reine Angew. Math. 410:1990;167-212.
5. N. D. Alikakos, P. W. Bates, X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc.
6. 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. 4:1991;903-910.
8. D. G. Aronson, Y. S. Huang, Limit and uniqueness of discrete rotating waves in large arrays Josephson junctions, Nonlinearity, 7, 1994, 777, 804.
9. Bates P. W., Fife P. C., Ren X., Wang X. Traveling waves in a convolution model for phase transitions. Arch. Rat. Mech. Anal. 138:1997;105-136.
10. P. W. Bates, K. Lu, C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc.
11. P. W. Bates, K. Lu, C. Zeng, Invariant foliations near normally hyperbolic invariant manifolds for semiflows, Trans. Amer. Math. Soc.
12. Bayliss A., Matkowsky B. J. Fronts, relaxation oscillations, and periodic doubling in solid fuel combustion. J. Comput. Phys. 71:1987;147-168.
13. Bell J. Some threshold results for models of myelinated nerves. Math. Biosci. 54:1981;181-190.
14. Bell J., Cosner C. Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons. Quart. Appl. Math. 42:1984;1-14.
15. Benzoni-Gavage S. Semi-discrete shock profiles for hyperbolic systems of conservation laws. Phys. D. 115:1998;109-123.
16. Berestycki H., Nicolaenko B., Scheurer B. Traveling wave solutions to combustion models and their singular limits. SIAM J. Math. Anal. 16:1985;1207-1242.
17. Cahn J. W. Theory of crystal growth and interface motion in crystalline materials. Acta Metallurgica. 8:1960;554-562.
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.
19. Chen X. Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Differential Equations. 2:1997;125-160.
20. Chow S.-N., Lin X.-B., Lu K. Smooth invariant foliations in infinite dimensional spaces. J. Differential Equations. 94:1991;266-291.
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. 42:1995;746-751.
22. Chow S.-N., Mallet-Paret J., Van Vleck E. S. Pattern formation and spatial chaos in spatially discrete evolution equations. Random Comput. Dynam. 4:1996;109-178.
23. Chow S.-N., Mallet-Paret J., Van Vleck E. S. Dynamics of lattice differential equations. Int. J. Bifur. Chaos Appl. Sci. Engrg. 6:1996;1605-1621.
24. Chow S.-N., Shen W. Dynamics in a discrete Nagumo equation: spatial topological chaos. SIAM J. Appl. Math. 55:1995;1764-1781.
25. Chow S.-N., Shen W. Stability and bifurcation of traveling wave solutions in coupled map lattices. Dynam. Systems Appl. 4:1995;1-25.
26. Chow S.-N., Shen W. A free boundary problem related to condensed two-phase combustion. Part I. Semigroup. J. Differential Equations. 108:1994;342-389.
27. Chow S.-N., Shen W. A free boundary problem related to condensed two-phase combustion. Part II. Stability and bifurcation. J. Differential Equations. 108:1994;390-423.
28. Chua L. O., Roska T. The CNN paradigm. IEEE Trans. Circuits Systems. 40:1993;147-156.
29. Chua L. O., Yang L. Cellular neural networks: Theory. IEEE Trans. Circuits Systems. 35:1988;1257-1272.
30. Chua L. O., Yang L. Cellular neural networks: Applications. IEEE Trans. Circuits Systems. 35:1988;1273-1290.
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. 33:1984;319-343.
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. 17:1969;765-773.
33. De Masi A., Gobron T., Presutti E. Traveling fronts in non-local evolution equations. Arch. Rational Mech. Anal. 132:1995;143-205.
34. 1. Bull. Acad. Polonaise Sci. 18:1970;27-33.
35. Elmer C. E., Van Vleck E. S. Computation of traveling waves for spatially discrete bistable reaction-diffusion equations. Appl. Numer. Math. 20:1996;157-169.
36. Ermentrout G. B. Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators. SIAM J. Appl. Math. 52:1992;1665-1687.
37. Ermentrout G. B., Kopell N. Inhibition-produced patterning in chains of coupled nonlinear oscillators. SIAM J. Appl. Math. 54:1994;478-507.
38. Ermentrout G. B., McLeod J. B. Existence and uniqueness of traveling waves for a neural network. Proc. Roy. Soc. Edinburg Sect. A. 123A:1993;461-478.
39. Erneux T., Nicolis G. Propagating waves in discrete bistable reaction diffusion systems. Phys. D. 67:1993;237-244.
40. Fenichel N. Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21:1971;193-226.
41. Fife P., McLeod J. B. The approach of solutions of nonlinear diffusion equations to traveling front solutions. Arch. Rational Mech. Anal. 65:1977;333-361.
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. 3:1998;85-110.
43. Firth W. J. Optical memory and spatial chaos. Phys. Rev. Lett. 61:1988;329-332.
44. Gardner R. Existence and stability of traveling wave solutions of competition models: a degree theoretic approach. J. Differential Equations. 44:1982;343-364.
45. Hale J. K. Ordinary Differential Equations. 1980;Krieger, Melbourne.
46. Hankerson D., Zinner B. Wavefronts for a cooperative tridiagonal system of differential equations. J. Dyn. Differential Equations. 5:1993;359-373.
47. Henry D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. 840:1981;Springer-Verlag, Berlin/New York.
48. Hillert M. A solid-solution model for inhomogeneous systems. Acta Metallurgica. 9:1961;525-535.
49. Kapral R. Discrete models for chemically reacting systems. J. Math. Chem. 6:1991;113-163.
50. Keener J. P. Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47:1987;556-572.
51. Keener J. P. The effects of discrete gap junction coupling on propagation in myocardium. J. Theor. Biol. 148:1991;49-82.
52. Kopell N., Ermentrout G. B., Williams T. L. On chains of oscillators forced at one end. SIAM J. Appl. Math. 51:1991;1397-1417.
53. Kopell N., Zhang W., Ermentrout G. B. Multiple coupling in chains of oscillators. SIAM J. Math. Anal. 21:1990;935-953.
54. Kuiper M. H. The homotopy type of the unitary group of Hilbert space. Topology. 3:1965;19-40.
55. Laplante J. P., Erneux T. Propagation failure in arrays of coupled bistable chemical reactors. J. Phys. Chem. 96:1992;4931-4934.
56. T.-P. Liu, S.-H. Yu, Continuum shock profiles for discrete conservation laws, I. Construction, Comm. Pure. Appl. Math.
57. Lorenz J. Numerics of invariant manifolds and attractors. Chaotic Numerics (Geelong, 1993). 1994;. p. 185-202.
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;337-346 World Scientific, Singapore.
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;105-129 North-Holland, Amsterdam.
60. J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dyn. Differential Equations.
61. J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dyn. Differential Equations.
62. Mallet-Paret J., Chow S.-N. Pattern formation and spatial chaos in lattice dynamical systems: II. IEEE Trans. Circuits Systems. 42:1995;752-756.
63. 0. Math. Ann. 174:1967;33-40.
64. Nishiura Y. Singular limit approach to stability and bifurcation for bistable reaction diffusion systems. Rocky Mountain J. Math. 21:1991;727-767.
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. 40:1993;872-877.
66. Pérez-Muñuzuri V., Pérez-Villar V., Chua L. O. Propagation failure in linear arrays of Chua's circuits. Int. J. Bifur. Chaos. 2:1992;403-406.
67. Sattinger D. H. On the stability of waves of nonlinear parabolic systems. Adv. Math. 22:1976;312-355.
68. Shen W. Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattices. SIAM J. Appl. Math. 56:1996;1379-1399.
69. Terman D. Traveling wave solutions arising from a two-step combustion model. SIAM J. Math. Anal. 19:1988;1057-1080.
70. Volpert A. I., Volpert V. A., Volpert V. A. Traveling Wave Solutions of Parabolic Systems. Translations of Mathematical Monographs. 140:1994;Amer. Math. Soc. Providence.
71. Weinberger H. F. Long-time behavior of a class of biological models. SIAM J. Math. Anal. 13:1982;353-396.
72. Winslow R. L., Kimball A. L., Varghese A. Simulating cardiac sinus and atrial network dynamics on the Connection Machine. Phys. D. 64:1993;281-298.
73. Xin X. Existence and uniqueness of travelling waves in a reaction-diffusion equation with combustion nonlinearity. Indiana Univ. Math. J. 40:1991;985-1008.
74. 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. 22:1991;1016-1020.
76. Zinner B. Existence of traveling wavefront solutions for the discrete Nagumo equation. J. Differential Equations. 96:1992;1-27.