American Institute of Mathematical Sciences

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June  2014, 1(2): 213-232. doi: 10.3934/jcd.2014.1.213

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, Norway
2. 

MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Szeged, Aradi vertanuk tere 1, H-6720, Hungary

Received  April 2013 Revised  July 2013 Published  December 2014

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: neural networks., Global stability, Neimark–Sacker bifurcation, rigorous numerics, strong resonance, graph representations, interval analysis.
Mathematics Subject Classification: 39A30, 39A28, 65Q10, 65G40, 92B2.
Citation: Ferenc A. Bartha, Ábel Garab. Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model. Journal of Computational Dynamics, 2014, 1 (2) : 213-232. doi: 10.3934/jcd.2014.1.213
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show all references

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Academic Press, New York, NY, 1983.  Google Scholar

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J. Difference Equ. Appl., 19 (2013), 2043-2078. doi: 10.1080/10236198.2013.804916.  Google Scholar

[3]

J. Differential Equations, 128 (1996), 46-57. doi: 10.1006/jdeq.1996.0088.  Google Scholar

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Physics Letters A, 358 (2006), 186-198. doi: 10.1016/j.physleta.2006.05.014.  Google Scholar

[5]

J. Math. Biol., 3 (1976), 381-391. doi: 10.1007/BF00275067.  Google Scholar

[6]

Numer. Math., 75 (1997), 293-317. doi: 10.1007/s002110050240.  Google Scholar

[7]

Chaos, 7 (1997), 221-228. doi: 10.1063/1.166223.  Google Scholar

[8]

J. Difference Equ. Appl., 11 (2005), 117-131. doi: 10.1080/10236190512331319334.  Google Scholar

[9]

Nonlinearity, 15 (2002), 1759-1779. doi: 10.1088/0951-7715/15/6/304.  Google Scholar

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Comput. Math. Appl., 47 (2004), 1249-1256. doi: 10.1016/S0898-1221(04)90119-8.  Google Scholar

[11]

Prentice-Hall, Upper Saddle River, NJ, 1999. Google Scholar

[12]

Proceedings of the Workshop Future Directions in Difference Equations, Colecc. Congr., Univ. Vigo, Serv. Publ., 69 (2011), 97-105. http://www.dma.uvigo.es/ eliz/pdf/Jimenez.pdf  Google Scholar

[13]

Mathematics and its Applications, vol. 256, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-017-1703-8.  Google Scholar

[14]

Electron. J. Qual. Theory Differ. Equ., Proc. 6'th Coll. Qualitative Theory of Diff. Equ., (1999), 1-12. http://www.math.u-szeged.hu/ejqtde/p76.pdf  Google Scholar

[15]

in Topics in Functional Differential and Difference Equations, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 29 (2001), 267-296. http://www.ams.org/bookstore?fn=20&arg1=ficseries&ikey=FIC-29  Google Scholar

[16]

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[21]

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[30]

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Princeton University Press, Princeton, NJ, 2011. http://press.princeton.edu/titles/9488.html  Google Scholar

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SIAM J. Appl. Dyn. Syst., 9 (2010), 1263-1283. doi: 10.1137/100795176.  Google Scholar

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de Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin, 2001. http://www.degruyter.com/view/product/61263 doi: 10.1515/9783110879971.  Google Scholar

[34]

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Computer-Aided Proofs in Analysis group, CAPA,, , ().   Google Scholar

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Computer Assisted Proofs in Dynamics group, CAPD Library,, , ().   Google Scholar

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National Information Infrastructure Development Institute, NIIF,, , ().   Google Scholar

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