Geodesic
Nonuniqueness of implicit lattice Nagumo equation
Applications of Mathematics, Tome 64 (2019) no. 2, pp. 169-194.

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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.
DOI : 10.21136/AM.2019.0270-18
Classification : 34A33, 35K57, 39A12, 65Q10
Keywords: reaction-diffusion equation; lattice differential equation; nonlinear algebraic problem; variational method; implicit discretization 
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Stehlík, Petr; Volek, Jonáš. Nonuniqueness of implicit lattice Nagumo equation. Applications of Mathematics, Tome 64 (2019) no. 2, pp. 169-194. doi : 10.21136/AM.2019.0270-18. http://geodesic.mathdoc.fr/articles/10.21136/AM.2019.0270-18/

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