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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