Journal of Computational Dynamics (JCD)

Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model

Pages: 213 - 232, Volume 1, Issue 2, December 2014      doi:10.3934/jcd.2014.1.213

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Ferenc A. Bartha - Department of Mathematical Sciences, NTNU, 7491 Trondheim, Norway (email)
Ábel Garab - MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Szeged, Aradi vertanuk tere 1, H-6720, Hungary (email)

Abstract: We consider the global asymptotic stability of the trivial fixed point of the difference equation $x_{n+1}=m x_n-\alpha \varphi(x_{n-1})$, where $(\alpha,m) \in \mathbb{R}^2$ and $\varphi$ is a real function satisfying the discrete Yorke condition: $\min\{0,x\} \leq \varphi(x) \leq \max\{0,x\}$ for all $x\in \mathbb{R}$. If $\varphi$ is bounded then $(\alpha,m) \in [|m|-1,1] \times [-1,1]$, $(\alpha,m) \neq (0,-1), (0,1)$ is necessary for the global stability of $0$. We prove that if $\varphi(x) \equiv \tanh(x)$, then this condition is sufficient as well.

Keywords:  Global stability, rigorous numerics, Neimark–Sacker bifurcation, strong resonance, graph representations, interval analysis, neural networks.
Mathematics Subject Classification:  39A30, 39A28, 65Q10, 65G40, 92B20.

Received: April 2013;      Revised: July 2013;      Available Online: December 2014.