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The Global Structure of Traveling Waves in Spatially Discrete Dynamical Systems

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Abstract

We obtain existence of traveling wave solutions for a class of spatially discrete systems, namely, lattice differential equations. Uniqueness of the wave speed c, and uniqueness of the solution with c≠0, are also shown. More generally, the global structure of the set of all traveling wave solutions is shown to be a smooth manifold where c≠0. Convergence results for solutions are obtained at the singular perturbation limit c → 0.

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REFERENCES

  1. Afraimovich, V. S., and Nekorkin, V. I. (1994). Chaos of traveling waves in a discrete chain of diffusively coupled maps. Int. J. Bifur. Chaos 4, 631–637.

    Google Scholar 

  2. Afraimovich, V. S., and Pesin, Ya. G. (1993). Traveling waves in lattice models of multidimensional and multi-component media, I. General hyperbolic properties. Nonlinearity 6, 429–455; (1993). II. Ergodic properties and dimension. Chaos 3, 233–241.

    Article  Google Scholar 

  3. Alikakos, N. D., Bates, P. W., and Chen, X. (in press). Periodic traveling waves and locating oscillating patterns in multidimensional domains. Trans. Amer. Math. Soc.

  4. Aronson, D. G. (1977). The asymptotic speed of propagation of a simple epidemic. In, Fitzgibbon, W. E., and Walker, H. F. (eds.), Nonlinear Diffusion, Pitman, pp. 1–23.

  5. Aronson, D. G., Golubitsky, M., and Mallet-Paret, J. (1991). Ponies on a merry-go-round in large arrays of Josephson junctions. Nonlinearity 4, 903–910.

    Article  Google Scholar 

  6. Aronson, D. G., and Huang, Y. S. (1994). Limit and uniqueness of discrete rotating waves in large arrays of Josephson junctions. Nonlinearity 7, 777–804.

    Article  Google Scholar 

  7. Aronson D. G., and Weinberger, H. F. (1978). Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76.

    Google Scholar 

  8. Atkinson, C., and Reuter, G. E. H. (1976). Deterministic epidemic waves. Math. Proc. Camb. Phil. Soc. 80, 315–330.

    Google Scholar 

  9. Bates, P. W., Fife, P. C., Ren, X., and Wang, X. (1997). Traveling waves in a convolution model for phase transitions. Arch. Rat. Mech. Anal. 138, 105–136.

    Article  Google Scholar 

  10. Bell, J. (1981). Some threshold results for models of myelinated nerves. Math. Biosci. 54, 181–190.

    Article  Google Scholar 

  11. Bell, J., and Cosner, C. (1984). Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons. Q. Appl. Math. 42, 1–14.

    Google Scholar 

  12. Benzoni-Gavage, S. (1998). Semi-discrete shock profiles for hyperbolic systems of conservation laws. Physica D 115, 109–123.

    Google Scholar 

  13. Cahn, J. W. (1960). Theory of crystal growth and interface motion in crystalline materials. Acta Metall. 8, 554–562.

    Article  Google Scholar 

  14. Cahn, J. W., Chow, S.-N., and Van Vleck, E. S. (1995). Spatially discrete nonlinear diffusion equations. Rocky Mount. J. Math. 25, 87–118.

    Google Scholar 

  15. Cahn, J. W., Mallet-Paret, J., and van Vleck, E. S. (in press). Traveling wave solutions for systems of ODE's on a two-dimensional spatial lattice. SIAM J. Appl. Math.

  16. Carpio, A., Chapman, S. J., Hastings, S. P., and McLeod, J. B. Wave solutions for a discrete reaction-diffusion equation, preprint.

  17. Chen, X. (1997). Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Diff. Eq. 2, 125–160.

    Google Scholar 

  18. Chow, S.-N., Diekmann, O., and Mallet-Paret, J. (1985). Stability, multiplicity, and global continuation of symmetric periodic solutions of a nonlinear Volterra integral equation. Jap. J. Appl. Math. 2, 433–469.

    Google Scholar 

  19. Chow, S.-N., Hale, J. K., and Mallet-Paret, J. (1980). An example of bifurcation to homoclinic orbits. J. Dig. Eq. 37, 351–373.

    Google Scholar 

  20. Chow, S.-N., Lin, X.-B., and Mallet-Paret, J. (1989). Transition layers for singularly perturbed delay differential equations with monotone nonlinearities. J. Dyn. Diff. Eq. 1, 3–43.

    Google Scholar 

  21. Chow, S.-N., and Mallet-Paret, J. (1995). Pattern formation and spatial chaos in lattice dynamical systems: I. IEEE Trans. Circuits Syst. 42, 746–751.

    Article  Google Scholar 

  22. Chow, S.-N., Mallet-Paret, J., and Shen, W. (in press). Traveling waves in lattice dynamical systems. J. Diff. Eq.

  23. Chow, S.-N., Mallet-Paret, J., and Van Vleck, E. S. (1996). Pattern formation and spatial chaos in spatially discrete evolution equations. Random Comp. Dyn. 4, 109–178.

    Google Scholar 

  24. Chow, S.-N., Mallet-Paret, J., and Van Vleck, E. S. Spatial entropy of stable mosaic solutions for a class of spatially discrete evolution equations, preprint.

  25. Chow, S.-N., Mallet-Paret, J., and Van Vleck, E. S. (1996). Dynamics of lattice differential equations. Int. J. Bifur. Chaos 6, 1605–1621.

    Article  Google Scholar 

  26. Chow, S.-N., and Shen, W. (1995). Stability and bifurcation of traveling wave solutions in coupled map lattices. J. Dyn. Syst. Appl. 4, 1–26.

    Google Scholar 

  27. Chua, L. O., and Roska, T. (1993). The CNN paradigm. IEEE Trans. Circuits Syst. 40, 147–156.

    Article  Google Scholar 

  28. Chua, L. O., and Yang, L. (1988). Cellular neural networks: Theory. IEEE Trans. Circuits Syst. 35, 1257–1272.

    Article  Google Scholar 

  29. Chua, L. O., and Yang, L. (1988). Cellular neural networks: Applications. IEEE Trans. Circuits Syst. 35, 1273–1290.

    Article  Google Scholar 

  30. Cook, H. E., de Fontaine, D., and Hilliard, J. E., (1969). A model for diffusion on cubic lattices and its application to the early stages of ordering. Acta Metall. 17, 765–773.

    Article  Google Scholar 

  31. De Masi, A., Gobron, T., and Presutti, E. (1995). Traveling fronts in non-local evolution equations. Arch. Rat. Mech. Anal. 132, 143–205.

    Google Scholar 

  32. Diekmann, O. (1978). Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol. 6, 109–130.

    Google Scholar 

  33. Diekmann, O. (1979). Run for your life. A note on the asymptotic speed of propagation of an epidemic. J. Diff. Eq. 33, 58–73.

    Google Scholar 

  34. Elmer C. E., and Van Vleck, E. S. (1996). Computation of traveling waves for spatially discrete bistable reaction-diffusion equations. Appl. Numer. Math. 20, 157–169.

    Article  Google Scholar 

  35. Elmer, C. E., and Van Vleck, E. S. Analysis and computation of traveling wave solutions of bistable differential-difference equations, preprint.

  36. Ermentrout, G. B. (1992). Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators. SIAM J. Appl. Math. 52, 1665–1687.

    Google Scholar 

  37. Ermentrout, G. B., and Kopell, N. (1994). Inhibition-produced patterning in chains of coupled nonlinear oscillators. SIAM J. Appl. Math. 54, 478–507.

    Google Scholar 

  38. Ermentrout, G. B., and McLeod, J. B. (1993). Existence and uniqueness of travelling waves for a neural network. Proc. Roy. Soc. Edinburgh 123A, 461–478.

    Google Scholar 

  39. Erneux, T., and Nicolis, G. (1993). Propagating waves in discrete bistable reaction-diffusion systems. Physica D 67, 237–244.

    Google Scholar 

  40. Fife, P., and McLeod, J. (1977). The approach of solutions of nonlinear diffusion equations to traveling front solutions. Arch. Rat. Mech. Anal. 65, 333–361.

    Google Scholar 

  41. Fife, P., and Wang, X. (1998). A convolution model for interfacial motion: The generation and propagation of internal layers in higher space dimensions. Adv. Diff. Eq. 3, 85–110.

    Google Scholar 

  42. Firth, W. J. (1988). Optical memory and spatial chaos. Phys. Rev. Lett. 61, 329–332.

    Article  Google Scholar 

  43. Fisher, R. J. (1937). The advance of advantageous genes. Ann. Eugenics 7, 355–369.

    Google Scholar 

  44. Hadeler, K.-P., and Zinner, B. Travelling fronts in discrete space and continuous time, preprint.

  45. Hale, J. K., and Lin, X.-B. (1986). Heteroclinic orbits for retarded functional differential equations. J. Diff. Eq. 65, 175–202.

    Google Scholar 

  46. Hale, J. K., and Verduyn Lunel, S. M. (1993). Introduction to Functional Differential Equations, Springer-Verlag, New York.

    Google Scholar 

  47. Hankerson, D., and Zinner, B. (1993). Wavefronts for a cooperative tridiagonal system of differential equations. J. Dyn. Diff. Eq. 5, 359–373.

    Google Scholar 

  48. Hillert, M. (1961). A solid-solution model for inhomogeneous systems. Acta Metall. 9, 525–535.

    Article  Google Scholar 

  49. Keener, J. P. (1987). Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47, 556–572.

    Google Scholar 

  50. Keener, J. P. (1991). The effects of discrete gap junction coupling on propagation in myocardium. J. Theor. Biol. 148, 49–82.

    Google Scholar 

  51. Kolmogoroff, A., Petrovsky, I., and Piscounoff, N. (1937). Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. Univ. État Moscou, Sér. Int. Sec. A 1:6, 1–25.

    Google Scholar 

  52. Kopell, N., and Ermentrout, G. B. (1990). Phase transitions and other phenomena in chains of coupled oscillators. SIAM J. Appl. Math. 50, 1014–1052.

    Google Scholar 

  53. Kopell, N., Ermentrout, G. B., and Williams, T. L. (1991). On chains of oscillators forced at one end. SIAM J. Appl. Math. 51, 1397–1417.

    Google Scholar 

  54. Kopell, N., Zhang, W., and Ermentrout, G. B. (1990). Multiple coupling in chains of oscillators. SIAM J. Math. Anal. 21, 935–953.

    Google Scholar 

  55. Laplante, J. P., and Erneux, T. (1992). Propagation failure in arrays of coupled bistable chemical reactors. J. Phys. Chem. 96, 4931–4934.

    Google Scholar 

  56. Levin, B. Ja. (1964). Distribution of Zeros of Entire Functions, Translations of Mathematical Monographs, Vol. 5, American Mathematical Society.

  57. Lin, X.-B. (1990). Using Mel'nikov's method to solve Šilnikov's problems. Proc. Roy. Soc. Edinburgh 116A, 295–325.

    Google Scholar 

  58. Lui, R. (1989). Biological growth and spread modeled by systems of recursions. I. Mathematical theory. Math. Biosci. 93, 269–295.

    Article  Google Scholar 

  59. Lui, R. (1989). Biological growth and spread modeled by systems of recursions. II. Biological theory. Math. Biosci. 93, 297–312.

    Article  Google Scholar 

  60. MacKay, R. S., and Sepulchre, J.-A. (1995). Multistability in networks of weakly coupled bistable units. Physica D 82, 243–254.

    Google Scholar 

  61. Mallet-Paret, J. (1996). Stability and oscillation in nonlinear cyclic systems. In Martelli, M., Cooke, K., Cumberbatch, E., Tang, B., and Thieme, H. (eds.), Differential Equations and Applications to Biology and Industry, World Scientific, pp. 337–346.

  62. Mallet-Paret, J. (1996). Spatial patterns, spatial chaos, and traveling waves in lattice differential equations. In van Strien, S. J., and Verduyn Lunel, S. M. (eds.), Stochastic and Spatial Structures of Dynamical Systems, North-Holland, Amsterdam, pp. 105–129.

    Google Scholar 

  63. Mallet-Paret, J. (1999). The Fredolm alternative for functional differential equations of mixed type. J. Dyn. Diff. Eq. 11, 1–47.

    Article  Google Scholar 

  64. Mallet-Paret, J., and Chow, S.-N. (1995). Pattern formation and spatial chaos in lattice dynamical systems: II. IEEE Trans. Circuits Syst. 42, 752–756.

    Article  Google Scholar 

  65. Mallet-Paret, J., and Sell, G. R. (1996). Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions. J. Diff. Eq. 125, 385–440.

    Google Scholar 

  66. Mallet-Paret, J., and Sell, G. R. (1996). The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay. J. Diff. Eq. 125, 441–489.

    Google Scholar 

  67. Mallet-Paret, J., and Smith, H. L. (1990). The Poincaré-Bendixon theorem for monotone cyclic feedback systems. J. Dyn. Diff. Eq. 2, 367–421.

    Google Scholar 

  68. Matthews, P. C., Mirollo, R. E., and Strogatz, S. H. Dynamics of a large system of coupled nonlinear oscillators. Physica D 52, 293–331.

  69. Mel'nikov, V. K. (1963). On the stability of the center for time periodic solutions. Trans. Moscow Math. Soc. 12, 3–56.

    Google Scholar 

  70. Mirollo, R. E., and Strogatz, S. H. (1990). Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math. 50, 1645–1662.

    Google Scholar 

  71. Pérez-Muñuzuri, A., Pérez-Muñuzuri, V., Pérez-Villar, V., and Chua, L. O. (1993). Spiral waves on a 2-d array of nonlinear circuits. IEEE Trans. Circuits Syst. 40, 872–877.

    Article  Google Scholar 

  72. Pérez-Muñuzuri, V., Pérez-Villar, V., and Chua, L. O. (1992). Propagation failure in linear arrays of Chua's circuits. Int. J. Bifur. Chaos 2, 403–406.

    Google Scholar 

  73. Peterhof, D., Sandstede, B., and Scheel, A. (1997). Exponential dichotomies for solitarywave solutions of semilinear elliptic equations on infinite cylinders. J. Diff. Eq. 140, 266–308.

    Google Scholar 

  74. Radcliffe, J., and Rass, L. (1983). Wave solutions for the deterministic non-reducible n-type epidemic. J. Math. Biol. 17, 45–66.

    Article  Google Scholar 

  75. Radcliffe, J., and Rass, L. (1984). The uniqueness of wave solutions for the deterministic non-reducible n-type epidemic. J. Math. Biol. 19, 303–308.

    Google Scholar 

  76. Radcliffe, J., and Rass, L. (1986). The asymptotic speed of propagation of the deterministic non-reducible retype epidemic. J. Math. Biol. 23, 341–359.

    Article  Google Scholar 

  77. Roska, T., and Chua, L. O. (1993). The CNN universal machine: an analogic array computer. IEEE Trans. Circuits Syst. 40, 163–173.

    Article  Google Scholar 

  78. Rudin, W. (1987). Real and Complex Analysis, McGraw-Hill, New York.

    Google Scholar 

  79. Rustichini, A. (1989). Functional differential equations of mixed type: The linear autonomous case. J. Dyn. Diff. Eq. 1, 121–143.

    Google Scholar 

  80. Rustichini, A. (1989). Hopf bifurcation for functional differential equations of mixed type. J. Dyn. Diff. Eq. 1, 145–177.

    Google Scholar 

  81. Sacker, R. J., and Sell, G. R. (1994). Dichotomies for linear evolution equations in Banach spaces. J. Diff. Eq. 113, 17–67.

    Google Scholar 

  82. Schumacher, K. (1980). Travelling-front solutions for integrodifferential equations II. In Jäger, W., Rost, H., and Tautu, P. (eds.), Biological Growth and Spread, Springer Lecture Notes in Biomathematics, Vol. 38, pp. 296–309.

  83. Shen, W. (1996). Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattice. SIAM J. Appl. Math. 56, 1379–1399.

    Google Scholar 

  84. Thieme, H. R. (1979). Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations. J. Reine Angew. Math. 306, 94–121.

    Google Scholar 

  85. Thieme, H. R. (1979). Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread. J. Math. Biol. 8, 173–187.

    Article  Google Scholar 

  86. Thiran, P., Crounse, K. R., Chua, L. O., and Hasler, M. (1995). Pattern formation properties of autonomous cellular neural networks. IEEE Trans. Cirucits Syst. 42, 757–774.

    Article  Google Scholar 

  87. Weinberger, H. F. (1980). Some deterministic models for the spread of genetic and other alterations. In Jäger, W., Rost, H., and Tautu, P. (eds.), Biological Growth and Spread, Springer Lecture Notes in Biomathematics, Vol. 38, pp. 320–349.

  88. Weinberger, H. F. (1982). Long time behavior of a class of biological models. SIAM J. Math. Anal. 13, 353–396.

    Google Scholar 

  89. Winslow, R. L., Kimball, A. L., and Varghese, A. (1993). Simulating cardiac sinus and atrial network dynamics on the Connection Machine. Physica D 64, 281–298.

    Google Scholar 

  90. Wu, J., and Xia, H. (1996). Self-sustained oscillations in a ring array of coupled lossless transmission lines. Appl. Math. Comp. 73, 55–76.

    Article  Google Scholar 

  91. Wu, J., and Zou, X. (1997). Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations. J. Diff. Eq. 135, 315–357.

    Google Scholar 

  92. Zinner, B (1991). Stability of traveling wavefronts for the discrete Nagumo equation. SIAM J. Math. Anal. 22, 1016–1020.

    Google Scholar 

  93. Zinner, B. (1992). Existence of traveling wavefront solutions for the discrete Nagumo equation. J. Diff. Eq. 96, 1–27.

    Google Scholar 

  94. Zinner, B., Harris, G., and Hudson, H. (1993). Traveling wavefronts for the discrete Fisher's equation. J. Diff. Eq. 105, 46–62.

    Google Scholar 

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  1. Division of Applied Mathematics, Brown University, Providence, Rhode Island, 02912

    John Mallet-Paret

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Mallet-Paret, J. The Global Structure of Traveling Waves in Spatially Discrete Dynamical Systems. Journal of Dynamics and Differential Equations 11, 49–127 (1999). https://doi.org/10.1023/A:1021841618074

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