Applications of Mathematics

, Volume 64, Issue 2, pp 169–194 | Cite as

Nonuniqueness of implicit lattice Nagumo equation

  • Petr StehlíkEmail author
  • Jonáš VolekEmail author


We consider the implicit discretization of Nagumo equation on finite lattices and show that its variational formulation corresponds in various parameter settings to convex, mountain-pass or saddle-point geometries. Consequently, we are able to derive conditions under which the implicit discretization yields multiple solutions. Interestingly, for certain parameters we show nonuniqueness for arbitrarily small discretization steps. Finally, we provide a simple example showing that the nonuniqueness can lead to complex dynamics in which the number of bounded solutions grows exponentially in time iterations, which in turn implies infinite number of global trajectories.


reaction-diffusion equation lattice differential equation nonlinear algebraic problem variational method implicit discretization 

MSC 2010

34A33 35K57 39A12 65Q10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors gratefully acknowledge discussions with Christian Pötzsche and Pavel Krejčí which led to the questions asked in this paper. They are also thankful for valuable suggestions of both referees and especially those of editor Tomáš Vejchodský.


  1. [1]
    G. Allaire, S. M. Kaber: Numerical Linear Algebra. Texts in Applied Mathematics 55, Springer, New York, 2008.Google Scholar
  2. [2]
    L. J. S. Allen: Persistence, extinction, and critical patch number for island populations. J. Math. Biol. 24 (1987), 617–625.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. Ambrosetti, P. H. Rabinowitz: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349–381.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    D. G. Aronson, H. F. Weinberger: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. Partial Differential Equations and Related Topics 1974. Lecture Notes in Mathematics 446, Springer, Berlin, 1975, pp. 5–49.CrossRefGoogle Scholar
  5. [5]
    S.-N. Chow, J. Mallet-Paret, W. Shen: Traveling waves in lattice dynamical systems. J. Differ. Equations 149 (1998), 248–291.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    S.-N. Chow, W. X. Shen: Dynamics in a discrete Nagumo equation: Spatial topological chaos. SIAM J. Appl. Math. 55 (1995), 1764–1781.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    L. O. Chua, L. Yang: Cellular neural networks: applications. IEEE Trans. Circuits Syst. 35 (1988), 1273–1290.MathSciNetCrossRefGoogle Scholar
  8. [8]
    D. C. Clark: A variant of the Lusternik-Schnirelman theory. Indiana Univ. Math. J. 22 (1972), 65–74.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    P. Drábek, J. Milota: Methods of Nonlinear Analysis. Applications to Differential Equations. Birkhäuser Advanced Texts Basler Lehrbücher, Springer, Basel, 2013.Google Scholar
  10. [10]
    P. C. Fife, J. B. McLeod: The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65 (1977), 335–361.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    M. Galewski, J. Smejda: On variational methods for nonlinear difference equations. J. Comput. Appl. Math. 233 (2010), 2985–2993.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    H. J. Hupkes, E. S. Van Vleck: Negative diffusion and traveling waves in high dimensional lattice systems. SIAM J. Math. Anal. 45 (2013), 1068–1135.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    H. J. Hupkes, E. S. Van Vleck: Travelling waves for complete discretizations of reaction diffusion systems. J. Dyn. Differ. Equations 28 (2016), 955–1006.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    J. P. Keener: Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47 (1987), 556–572.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    J. Mallet-Paret: The global structure of traveling waves in spatially discrete dynamical systems. J. Dyn. Differ. Equations 11 (1999), 49–127.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    G. Molica Bisci, D. Repovš: On some variational algebraic problems. Adv. Nonlinear Anal. 2 (2013), 127–146.MathSciNetzbMATHGoogle Scholar
  17. [17]
    J. Nagumo, S. Arimoto, S. Yoshizawa: An active pulse transmission line simulating nerve axon. Proc. IRE 50 (1962), 2061–2070.CrossRefGoogle Scholar
  18. [18]
    J. Otta, P. Stehlík: Multiplicity of solutions for discrete problems with double-well potentials. Electron. J. Differ. Equ. 2013 (2013), 14 pages.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    C. Pötzsche: Geometric Theory of Discrete Nonautonomous Dynamical Systems. Lecture Notes in Mathematics 2002, Springer, Berlin, 2010.CrossRefzbMATHGoogle Scholar
  20. [20]
    P. H. Rabinowitz: Some minimax theorems and applications to nonlinear partial differential equations. Nonlinear Analysis. Academic Press, New York, 1978, pp. 161–177.Google Scholar
  21. [21]
    P. H. Rabinowitz: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics 65, American Mathematical Society, Providence, 1986.CrossRefzbMATHGoogle Scholar
  22. [22]
    A. Slavík: Invariant regions for systems of lattice reaction-diffusion equations. J. Differ. Equations 263 (2017), 7601–7626.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    A. Slavík, P. Stehlík: Dynamic diffusion-type equations on discrete-space domains. J. Math. Anal. Appl. 427 (2015), 525–545.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    P. Stehlík: Exponential number of stationary solutions for Nagumo equations on graphs. J. Math. Anal. Appl. 455 (2017), 1749–1764.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    P. Stehlík, J. Volek: Maximum principles for discrete and semidiscrete reaction-diffusion equation. Discrete Dyn. Nat. Soc. 2015 (2015), Article ID 791304, 13 pages.MathSciNetCrossRefGoogle Scholar
  26. [26]
    P. Stehlík, J. Volek: Variational methods and implicit discrete Nagumo equation. J. Math. Anal. Appl. 438 (2016), 643–656.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    J. Volek: Landesman-Lazer conditions for difference equations involving sublinear perturbations. J. Difference Equ. Appl. 22 (2016), 1698–1719.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    J. Volek: Multiple critical points of saddle geometry functionals. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 170 (2018), 238–257.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    B. Zinner: Existence of traveling wavefront solutions for the discrete Nagumo equation. J. Differ. Equations 96 (1992), 1–27.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Cz 2019

Authors and Affiliations

  1. 1.Department of Mathematics and New Technologies for the Information SocietyUniversity of West BohemiaPlzeňCzech Republic

Personalised recommendations