# American Institute of Mathematical Sciences

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

 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.
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:
 [1] G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, New York, NY, 1983. [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. [3] Y. Cao, Uniqueness of periodic solution for differential delay equations, J. Differential Equations, 128 (1996), 46-57. doi: 10.1006/jdeq.1996.0088. [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. [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. [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. [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. [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. [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. [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. [11] S. Haykin, Neural Networks: A Comprehensive Foundation, Prentice-Hall, Upper Saddle River, NJ, 1999. [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 [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. [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 [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 [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. [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 [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. [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. [20] S. Luzzatto and P. Pilarczyk, Finite resolution dynamics, Found. Comput. Math., 11 (2011), 211-239. doi: 10.1007/s10208-010-9083-z. [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. [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. [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. [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. [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. [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. [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. [28] J. G. Siek, L.-Q. Lee and A. Lumsdaine, The Boost Graph Library: User Guide and Reference Manual, Addison-Wesley, Boston, MA, 2002. [29] R. Tarjan, Depth-first search and linear graph algorithms, SIAM J. Comput., 1 (1972), 146-160. doi: 10.1137/0201010. [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. [31] W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations, Princeton University Press, Princeton, NJ, 2011. http://press.princeton.edu/titles/9488.html [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. [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. [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. [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. [36] [37] Computer-Aided Proofs in Analysis group, CAPA,, , (). [38] Computer Assisted Proofs in Dynamics group, CAPD Library,, , (). [39] National Information Infrastructure Development Institute, NIIF,, , ().

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##### References:
 [1] G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, New York, NY, 1983. [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. [3] Y. Cao, Uniqueness of periodic solution for differential delay equations, J. Differential Equations, 128 (1996), 46-57. doi: 10.1006/jdeq.1996.0088. [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. [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. [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. [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. [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. [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. [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. [11] S. Haykin, Neural Networks: A Comprehensive Foundation, Prentice-Hall, Upper Saddle River, NJ, 1999. [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 [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. [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 [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 [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. [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 [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. [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. [20] S. Luzzatto and P. Pilarczyk, Finite resolution dynamics, Found. Comput. Math., 11 (2011), 211-239. doi: 10.1007/s10208-010-9083-z. [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. [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. [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. [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. [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. [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. [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. [28] J. G. Siek, L.-Q. Lee and A. Lumsdaine, The Boost Graph Library: User Guide and Reference Manual, Addison-Wesley, Boston, MA, 2002. [29] R. Tarjan, Depth-first search and linear graph algorithms, SIAM J. Comput., 1 (1972), 146-160. doi: 10.1137/0201010. [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. [31] W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations, Princeton University Press, Princeton, NJ, 2011. http://press.princeton.edu/titles/9488.html [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. [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. [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. [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. [36] [37] Computer-Aided Proofs in Analysis group, CAPA,, , (). [38] Computer Assisted Proofs in Dynamics group, CAPD Library,, , (). [39] National Information Infrastructure Development Institute, NIIF,, , ().
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