Abstract
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.
Similar content being viewed by others
References
G. Allaire, S. M. Kaber: Numerical Linear Algebra. Texts in Applied Mathematics 55, Springer, New York, 2008.
L. J. S. Allen: Persistence, extinction, and critical patch number for island populations. J. Math. Biol. 24 (1987), 617–625.
A. Ambrosetti, P. H. Rabinowitz: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349–381.
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.
S.-N. Chow, J. Mallet-Paret, W. Shen: Traveling waves in lattice dynamical systems. J. Differ. Equations 149 (1998), 248–291.
S.-N. Chow, W. X. Shen: Dynamics in a discrete Nagumo equation: Spatial topological chaos. SIAM J. Appl. Math. 55 (1995), 1764–1781.
L. O. Chua, L. Yang: Cellular neural networks: applications. IEEE Trans. Circuits Syst. 35 (1988), 1273–1290.
D. C. Clark: A variant of the Lusternik-Schnirelman theory. Indiana Univ. Math. J. 22 (1972), 65–74.
P. Drábek, J. Milota: Methods of Nonlinear Analysis. Applications to Differential Equations. Birkhäuser Advanced Texts Basler Lehrbücher, Springer, Basel, 2013.
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.
M. Galewski, J. Smejda: On variational methods for nonlinear difference equations. J. Comput. Appl. Math. 233 (2010), 2985–2993.
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.
H. J. Hupkes, E. S. Van Vleck: Travelling waves for complete discretizations of reaction diffusion systems. J. Dyn. Differ. Equations 28 (2016), 955–1006.
J. P. Keener: Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47 (1987), 556–572.
J. Mallet-Paret: The global structure of traveling waves in spatially discrete dynamical systems. J. Dyn. Differ. Equations 11 (1999), 49–127.
G. Molica Bisci, D. Repovš: On some variational algebraic problems. Adv. Nonlinear Anal. 2 (2013), 127–146.
J. Nagumo, S. Arimoto, S. Yoshizawa: An active pulse transmission line simulating nerve axon. Proc. IRE 50 (1962), 2061–2070.
J. Otta, P. Stehlík: Multiplicity of solutions for discrete problems with double-well potentials. Electron. J. Differ. Equ. 2013 (2013), 14 pages.
C. Pötzsche: Geometric Theory of Discrete Nonautonomous Dynamical Systems. Lecture Notes in Mathematics 2002, Springer, Berlin, 2010.
P. H. Rabinowitz: Some minimax theorems and applications to nonlinear partial differential equations. Nonlinear Analysis. Academic Press, New York, 1978, pp. 161–177.
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.
A. Slavík: Invariant regions for systems of lattice reaction-diffusion equations. J. Differ. Equations 263 (2017), 7601–7626.
A. Slavík, P. Stehlík: Dynamic diffusion-type equations on discrete-space domains. J. Math. Anal. Appl. 427 (2015), 525–545.
P. Stehlík: Exponential number of stationary solutions for Nagumo equations on graphs. J. Math. Anal. Appl. 455 (2017), 1749–1764.
P. Stehlík, J. Volek: Maximum principles for discrete and semidiscrete reaction-diffusion equation. Discrete Dyn. Nat. Soc. 2015 (2015), Article ID 791304, 13 pages.
P. Stehlík, J. Volek: Variational methods and implicit discrete Nagumo equation. J. Math. Anal. Appl. 438 (2016), 643–656.
J. Volek: Landesman-Lazer conditions for difference equations involving sublinear perturbations. J. Difference Equ. Appl. 22 (2016), 1698–1719.
J. Volek: Multiple critical points of saddle geometry functionals. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 170 (2018), 238–257.
B. Zinner: Existence of traveling wavefront solutions for the discrete Nagumo equation. J. Differ. Equations 96 (1992), 1–27.
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding authors
Additional information
This publication was supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports under the program NPU I.
Rights and permissions
About this article
Cite this article
Stehlík, P., Volek, J. Nonuniqueness of implicit lattice Nagumo equation. Appl Math 64, 169–194 (2019). https://doi.org/10.21136/AM.2019.0270-18
Received:
Published:
Issue Date:
DOI: https://doi.org/10.21136/AM.2019.0270-18
Keywords
- reaction-diffusion equation
- lattice differential equation
- nonlinear algebraic problem
- variational method
- implicit discretization