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
References:
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G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, New York, NY, 1983.  Google Scholar

[2]

F. A. Bartha, Á. Garab and T. Krisztin, Local stability implies global stability for the 2-dimensional Ricker map, J. Difference Equ. Appl., 19 (2013), 2043-2078. doi: 10.1080/10236198.2013.804916.  Google Scholar

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M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numer. Math., 75 (1997), 293-317. doi: 10.1007/s002110050240.  Google Scholar

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M. Dellnitz, A. Hohmann, O. Junge and M. Rumpf, Exploring invariant sets and invariant measures, Chaos, 7 (1997), 221-228. doi: 10.1063/1.166223.  Google Scholar

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H. A. El-Morshedy and E. Liz, Convergence to equilibria in discrete population models, J. Difference Equ. Appl., 11 (2005), 117-131. doi: 10.1080/10236190512331319334.  Google Scholar

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S. Guo, L. Huang and L. Wang, Exponential stability of discrete-time Hopfield neural networks, Comput. Math. Appl., 47 (2004), 1249-1256. doi: 10.1016/S0898-1221(04)90119-8.  Google Scholar

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S. Haykin, Neural Networks: A Comprehensive Foundation, Prentice-Hall, Upper Saddle River, NJ, 1999. Google Scholar

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V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and its Applications, vol. 256, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-017-1703-8.  Google Scholar

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T. Krisztin, Periodic orbits and the global attractor for delayed monotone negative feedback, 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

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[17]

T. Krisztin, H.-O. Walther and J. Wu, Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback, Fields Institute Monographs, vol. 11, American Mathematical Society, Providence, RI, 1999. http://www.ams.org/bookstore?fn=20&arg1=fimseries&ikey=FIM-11  Google Scholar

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E. Liz, Stability of non-autonomous difference equations: Simple ideas leading to useful results, J. Difference Equ. Appl., 17 (2011), 203-220. doi: 10.1080/10236198.2010.549007.  Google Scholar

[20]

S. Luzzatto and P. Pilarczyk, Finite resolution dynamics, Found. Comput. Math., 11 (2011), 211-239. doi: 10.1007/s10208-010-9083-z.  Google Scholar

[21]

S. Mohamad and K. Gopalsamy, Exponential stability of continuous-time and discrete-time cellular neural networks with delays, Appl. Math. Comput., 135 (2003), 17-38. doi: 10.1016/S0096-3003(01)00299-5.  Google Scholar

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R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, SIAM Soc. for Industrial and Applied Math., Philadelphia, PA, 2009. doi: 10.1137/1.9780898717716.  Google Scholar

[24]

N. S. Nedialkov, K. R. Jackson and G. F. Corliss, Validated solutions of initial value problems for ordinary differential equations, Appl. Math. Comput., 105 (1999), 21-68. doi: 10.1016/S0096-3003(98)10083-8.  Google Scholar

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O. I. Nenya, On the global stability of one nonlinear difference equation, Nonlinear Oscil., 9 (2006), 513-522. doi: 10.1007/s11072-006-0058-6.  Google Scholar

[26]

O. I. Nenya, V. I. Tkachenko and S. I. Trofimchuk, On sharp conditions for the global stability of a difference equation satisfying the Yorke condition, Ukrainian Math. J., 60 (2008), 78-90. doi: 10.1007/s11253-008-0043-6.  Google Scholar

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O. I. Nenya, V. I. Tkachenko and S. I. Trofimchuk, On the global stability of one nonlinear difference equation, Nonlinear Oscil., 7 (2004), 473-480. doi: 10.1007/s11072-005-0027-5.  Google Scholar

[28]

J. G. Siek, L.-Q. Lee and A. Lumsdaine, The Boost Graph Library: User Guide and Reference Manual, Addison-Wesley, Boston, MA, 2002. Google Scholar

[29]

R. Tarjan, Depth-first search and linear graph algorithms, SIAM J. Comput., 1 (1972), 146-160. doi: 10.1137/0201010.  Google Scholar

[30]

W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math., 2 (2002), 53-117. http://www2.math.uu.se/ warwick/main/rodes/JFoCM.pdf doi: 10.1007/s002080010018.  Google Scholar

[31]

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[32]

D. Wilczak, Uniformly hyperbolic attractor of the Smale-Williams type for a Poincaré map in the Kuznetsov system, SIAM J. Appl. Dyn. Syst., 9 (2010), 1263-1283. doi: 10.1137/100795176.  Google Scholar

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J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, 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

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[35]

Q. Zhang, X. Wei and J. Xu, On global exponential stability of discrete-time Hopfield neural networks with variable delays, Discrete Dyn. Nat. Soc., 2007 (2007), Art. ID 67675, 9pp. doi: 10.1155/2007/67675.  Google Scholar

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, =, ().   Google Scholar

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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

show all references

References:
[1]

G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, New York, NY, 1983.  Google Scholar

[2]

F. A. Bartha, Á. Garab and T. Krisztin, Local stability implies global stability for the 2-dimensional Ricker map, J. Difference Equ. Appl., 19 (2013), 2043-2078. doi: 10.1080/10236198.2013.804916.  Google Scholar

[3]

Y. Cao, Uniqueness of periodic solution for differential delay equations, J. Differential Equations, 128 (1996), 46-57. doi: 10.1006/jdeq.1996.0088.  Google Scholar

[4]

W.-H. Chen, X. Lu and D.-Y. Liang, Global exponential stability for discrete-time neural networks with variable delays, Physics Letters A, 358 (2006), 186-198. doi: 10.1016/j.physleta.2006.05.014.  Google Scholar

[5]

C. W. Clark, A delayed-recruitment model of population dynamics, with an application to baleen whale populations, J. Math. Biol., 3 (1976), 381-391. doi: 10.1007/BF00275067.  Google Scholar

[6]

M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numer. Math., 75 (1997), 293-317. doi: 10.1007/s002110050240.  Google Scholar

[7]

M. Dellnitz, A. Hohmann, O. Junge and M. Rumpf, Exploring invariant sets and invariant measures, Chaos, 7 (1997), 221-228. doi: 10.1063/1.166223.  Google Scholar

[8]

H. A. El-Morshedy and E. Liz, Convergence to equilibria in discrete population models, J. Difference Equ. Appl., 11 (2005), 117-131. doi: 10.1080/10236190512331319334.  Google Scholar

[9]

Z. Galias, Rigorous investigation of the Ikeda map by means of interval arithmetic, Nonlinearity, 15 (2002), 1759-1779. doi: 10.1088/0951-7715/15/6/304.  Google Scholar

[10]

S. Guo, L. Huang and L. Wang, Exponential stability of discrete-time Hopfield neural networks, Comput. Math. Appl., 47 (2004), 1249-1256. doi: 10.1016/S0898-1221(04)90119-8.  Google Scholar

[11]

S. Haykin, Neural Networks: A Comprehensive Foundation, Prentice-Hall, Upper Saddle River, NJ, 1999. Google Scholar

[12]

V. J. López, A counterexample on global attractivity for Clark's equation, 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]

V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and its Applications, vol. 256, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-017-1703-8.  Google Scholar

[14]

T. Krisztin, Periodic orbits and the global attractor for delayed monotone negative feedback, 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]

T. Krisztin, Unstable sets of periodic orbits and the global attractor for delayed feedback, 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]

T. Krisztin and H.-O. Walther, Unique periodic orbits for delayed positive feedback and the global attractor, J. Dynam. Differential Equations, 13 (2001), 1-57. doi: 10.1023/A:1009091930589.  Google Scholar

[17]

T. Krisztin, H.-O. Walther and J. Wu, Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback, Fields Institute Monographs, vol. 11, American Mathematical Society, Providence, RI, 1999. http://www.ams.org/bookstore?fn=20&arg1=fimseries&ikey=FIM-11  Google Scholar

[18]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd edition, Applied Mathematical Sciences, vol. 112, Springer-Verlag, New York, NY, 2004. doi: 10.1007/978-1-4757-3978-7.  Google Scholar

[19]

E. Liz, Stability of non-autonomous difference equations: Simple ideas leading to useful results, J. Difference Equ. Appl., 17 (2011), 203-220. doi: 10.1080/10236198.2010.549007.  Google Scholar

[20]

S. Luzzatto and P. Pilarczyk, Finite resolution dynamics, Found. Comput. Math., 11 (2011), 211-239. doi: 10.1007/s10208-010-9083-z.  Google Scholar

[21]

S. Mohamad and K. Gopalsamy, Exponential stability of continuous-time and discrete-time cellular neural networks with delays, Appl. Math. Comput., 135 (2003), 17-38. doi: 10.1016/S0096-3003(01)00299-5.  Google Scholar

[22]

R. E. Moore, Methods and Applications of Interval Analysis, SIAM Soc. for Industrial and Applied Math., Philadelphia, PA, 1979. doi: 10.1137/1.9781611970906.  Google Scholar

[23]

R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, SIAM Soc. for Industrial and Applied Math., Philadelphia, PA, 2009. doi: 10.1137/1.9780898717716.  Google Scholar

[24]

N. S. Nedialkov, K. R. Jackson and G. F. Corliss, Validated solutions of initial value problems for ordinary differential equations, Appl. Math. Comput., 105 (1999), 21-68. doi: 10.1016/S0096-3003(98)10083-8.  Google Scholar

[25]

O. I. Nenya, On the global stability of one nonlinear difference equation, Nonlinear Oscil., 9 (2006), 513-522. doi: 10.1007/s11072-006-0058-6.  Google Scholar

[26]

O. I. Nenya, V. I. Tkachenko and S. I. Trofimchuk, On sharp conditions for the global stability of a difference equation satisfying the Yorke condition, Ukrainian Math. J., 60 (2008), 78-90. doi: 10.1007/s11253-008-0043-6.  Google Scholar

[27]

O. I. Nenya, V. I. Tkachenko and S. I. Trofimchuk, On the global stability of one nonlinear difference equation, Nonlinear Oscil., 7 (2004), 473-480. doi: 10.1007/s11072-005-0027-5.  Google Scholar

[28]

J. G. Siek, L.-Q. Lee and A. Lumsdaine, The Boost Graph Library: User Guide and Reference Manual, Addison-Wesley, Boston, MA, 2002. Google Scholar

[29]

R. Tarjan, Depth-first search and linear graph algorithms, SIAM J. Comput., 1 (1972), 146-160. doi: 10.1137/0201010.  Google Scholar

[30]

W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math., 2 (2002), 53-117. http://www2.math.uu.se/ warwick/main/rodes/JFoCM.pdf doi: 10.1007/s002080010018.  Google Scholar

[31]

W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations, Princeton University Press, Princeton, NJ, 2011. http://press.princeton.edu/titles/9488.html  Google Scholar

[32]

D. Wilczak, Uniformly hyperbolic attractor of the Smale-Williams type for a Poincaré map in the Kuznetsov system, SIAM J. Appl. Dyn. Syst., 9 (2010), 1263-1283. doi: 10.1137/100795176.  Google Scholar

[33]

J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, 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]

H. Xu, Y. Chen and K. L. Teo, Global exponential stability of impulsive discrete-time neural networks with time-varying delays, Appl. Math. Comput., 217 (2010), 537-544. doi: 10.1016/j.amc.2010.05.087.  Google Scholar

[35]

Q. Zhang, X. Wei and J. Xu, On global exponential stability of discrete-time Hopfield neural networks with variable delays, Discrete Dyn. Nat. Soc., 2007 (2007), Art. ID 67675, 9pp. doi: 10.1155/2007/67675.  Google Scholar

[36]

, =, ().   Google Scholar

[37]

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

[38]

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

[39]

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

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