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

Ferenc A. Bartha; Ábel Garab

doi:10.3934/jcd.2014.1.213

Article Contents

2014, Volume 1, Issue 2: 213-232. Doi: 10.3934/jcd.2014.1.213

This issue Previous Article Next Article Reconstructing functions from random samples

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

Ferenc A. Bartha^1, and
Ábel Garab^2,

1.
Department of Mathematical Sciences, NTNU, 7491 Trondheim
2.
MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Szeged, Aradi vertanuk tere 1, H-6720

Received: March 31, 2013

Revised: June 30, 2013

Published: May 2014

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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.

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Mathematics Subject Classification: 39A30, 39A28, 65Q10, 65G40, 92B20.

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