Nonuniqueness of implicit lattice Nagumo equation

Petr Stehlík; Jonáš Volek

Applications of Mathematics (2019)

  • Volume: 64, Issue: 2, page 169-194
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Stehlík, Petr, and Volek, Jonáš. "Nonuniqueness of implicit lattice Nagumo equation." Applications of Mathematics 64.2 (2019): 169-194. <http://eudml.org/doc/294225>.

@article{Stehlík2019,
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.},
author = {Stehlík, Petr, Volek, Jonáš},
journal = {Applications of Mathematics},
keywords = {reaction-diffusion equation; lattice differential equation; nonlinear algebraic problem; variational method; implicit discretization},
language = {eng},
number = {2},
pages = {169-194},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Nonuniqueness of implicit lattice Nagumo equation},
url = {http://eudml.org/doc/294225},
volume = {64},
year = {2019},
}

TY - JOUR
AU - Stehlík, Petr
AU - Volek, Jonáš
TI - Nonuniqueness of implicit lattice Nagumo equation
JO - Applications of Mathematics
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 2
SP - 169
EP - 194
AB - 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.
LA - eng
KW - reaction-diffusion equation; lattice differential equation; nonlinear algebraic problem; variational method; implicit discretization
UR - http://eudml.org/doc/294225
ER -

References

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  1. Allaire, G., Kaber, S. M., 10.1007/978-0-387-68918-0, Texts in Applied Mathematics 55, Springer, New York (2008). (2008) Zbl1135.65014MR2365296DOI10.1007/978-0-387-68918-0
  2. Allen, L. J. S., 10.1007/BF00275506, J. Math. Biol. 24 (1987), 617-625. (1987) Zbl0603.92019MR0880448DOI10.1007/BF00275506
  3. Ambrosetti, A., Rabinowitz, P. H., 10.1016/0022-1236(73)90051-7, J. Funct. Anal. 14 (1973), 349-381. (1973) Zbl0273.49063MR0370183DOI10.1016/0022-1236(73)90051-7
  4. Aronson, D. G., Weinberger, H. F., 10.1007/bfb0070595, Partial Differential Equations and Related Topics 1974 Lecture Notes in Mathematics 446, Springer, Berlin (1975), 5-49. (1975) Zbl0325.35050MR0427837DOI10.1007/bfb0070595
  5. Chow, S.-N., Mallet-Paret, J., Shen, W., 10.1006/jdeq.1998.3478, J. Differ. Equations 149 (1998), 248-291. (1998) Zbl0911.34050MR1646240DOI10.1006/jdeq.1998.3478
  6. Chow, S.-N., Shen, W. X., 10.1137/S0036139994261757, SIAM J. Appl. Math. 55 (1995), 1764-1781. (1995) Zbl0840.34012MR1358800DOI10.1137/S0036139994261757
  7. Chua, L. O., Yang, L., 10.1109/31.7601, IEEE Trans. Circuits Syst. 35 (1988), 1273-1290. (1988) MR0960778DOI10.1109/31.7601
  8. Clark, D. C., 10.1512/iumj.1972.22.22008, Indiana Univ. Math. J. 22 (1972), 65-74. (1972) Zbl0228.58006MR0296777DOI10.1512/iumj.1972.22.22008
  9. Drábek, P., Milota, J., 10.1007/978-3-0348-0387-8, Birkhäuser Advanced Texts Basler Lehrbücher, Springer, Basel (2013). (2013) Zbl1264.35001MR3025694DOI10.1007/978-3-0348-0387-8
  10. Fife, P. C., McLeod, J. B., 10.1007/BF00250432, Arch. Ration. Mech. Anal. 65 (1977), 335-361. (1977) Zbl0361.35035MR0442480DOI10.1007/BF00250432
  11. Galewski, M., Smejda, J., 10.1016/j.cam.2009.11.044, J. Comput. Appl. Math. 233 (2010), 2985-2993. (2010) Zbl1187.39006MR2592272DOI10.1016/j.cam.2009.11.044
  12. Hupkes, H. J., Vleck, E. S. Van, 10.1137/120880628, SIAM J. Math. Anal. 45 (2013), 1068-1135. (2013) Zbl1301.34098MR3049651DOI10.1137/120880628
  13. Hupkes, H. J., Vleck, E. S. Van, 10.1007/s10884-014-9423-9, J. Dyn. Differ. Equations 28 (2016), 955-1006. (2016) Zbl1353.34094MR3537361DOI10.1007/s10884-014-9423-9
  14. Keener, J. P., 10.1137/0147038, SIAM J. Appl. Math. 47 (1987), 556-572. (1987) Zbl0649.34019MR0889639DOI10.1137/0147038
  15. Mallet-Paret, J., 10.1023/A:1021841618074, J. Dyn. Differ. Equations 11 (1999), 49-127. (1999) Zbl0921.34046MR1680459DOI10.1023/A:1021841618074
  16. Bisci, G. Molica, Repovš, D., 10.1515/anona-2012-0028, Adv. Nonlinear Anal. 2 (2013), 127-146. (2013) Zbl1273.39003MR3055530DOI10.1515/anona-2012-0028
  17. Nagumo, J., Arimoto, S., Yoshizawa, S., 10.1109/jrproc.1962.288235, Proc. IRE 50 (1962), 2061-2070. (1962) DOI10.1109/jrproc.1962.288235
  18. Otta, J., Stehlík, P., Multiplicity of solutions for discrete problems with double-well potentials, Electron. J. Differ. Equ. 2013 (2013), 14 pages. (2013) Zbl1287.39008MR3104962
  19. Pötzsche, C., 10.1007/978-3-642-14258-1, Lecture Notes in Mathematics 2002, Springer, Berlin (2010). (2010) Zbl1247.37003MR2680867DOI10.1007/978-3-642-14258-1
  20. Rabinowitz, P. H., 10.1016/b978-0-12-165550-1.50016-1, Nonlinear Analysis Academic Press, New York (1978), 161-177. (1978) Zbl0466.58015MR0501092DOI10.1016/b978-0-12-165550-1.50016-1
  21. Rabinowitz, P. H., 10.1090/cbms/065, CBMS Regional Conference Series in Mathematics 65, American Mathematical Society, Providence (1986). (1986) Zbl0609.58002MR0845785DOI10.1090/cbms/065
  22. Slavík, A., 10.1016/j.jde.2017.08.019, J. Differ. Equations 263 (2017), 7601-7626. (2017) Zbl06782926MR3705693DOI10.1016/j.jde.2017.08.019
  23. Slavík, A., Stehlík, P., 10.1016/j.jmaa.2015.02.056, J. Math. Anal. Appl. 427 (2015), 525-545. (2015) Zbl1338.35449MR3318214DOI10.1016/j.jmaa.2015.02.056
  24. Stehlík, P., 10.1016/j.jmaa.2017.06.075, J. Math. Anal. Appl. 455 (2017), 1749-1764. (2017) Zbl06759300MR3671252DOI10.1016/j.jmaa.2017.06.075
  25. Stehlík, P., Volek, J., 10.1155/2015/791304, Discrete Dyn. Nat. Soc. 2015 (2015), Article ID 791304, 13 pages. (2015) MR3407064DOI10.1155/2015/791304
  26. Stehlík, P., Volek, J., 10.1016/j.jmaa.2016.02.009, J. Math. Anal. Appl. 438 (2016), 643-656. (2016) Zbl1334.35134MR3466056DOI10.1016/j.jmaa.2016.02.009
  27. Volek, J., 10.1080/10236198.2016.1234617, J. Difference Equ. Appl. 22 (2016), 1698-1719. (2016) Zbl1361.39003MR3590409DOI10.1080/10236198.2016.1234617
  28. Volek, J., 10.1016/j.na.2018.01.008, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 170 (2018), 238-257. (2018) Zbl1386.49006MR3765563DOI10.1016/j.na.2018.01.008
  29. Zinner, B., 10.1016/0022-0396(92)90142-A, J. Differ. Equations 96 (1992), 1-27. (1992) Zbl0752.34007MR1153307DOI10.1016/0022-0396(92)90142-A

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