# 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, (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. doi: 10.1080/10236198.2013.804916. [3] Y. Cao, Uniqueness of periodic solution for differential delay equations,, J. Differential Equations, 128 (1996), 46. 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. 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. 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. doi: 10.1007/s002110050240. [7] M. Dellnitz, A. Hohmann, O. Junge and M. Rumpf, Exploring invariant sets and invariant measures,, Chaos, 7 (1997), 221. 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. doi: 10.1080/10236190512331319334. [9] Z. Galias, Rigorous investigation of the Ikeda map by means of interval arithmetic,, Nonlinearity, 15 (2002), 1759. 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. doi: 10.1016/S0898-1221(04)90119-8. [11] S. Haykin, Neural Networks: A Comprehensive Foundation,, Prentice-Hall, (1999). [12] V. J. López, A counterexample on global attractivity for Clark's equation,, Proceedings of the Workshop Future Directions in Difference Equations, 69 (2011), 97. [13] V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications,, Mathematics and its Applications, (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., (1999), 1. [15] T. Krisztin, Unstable sets of periodic orbits and the global attractor for delayed feedback,, in Topics in Functional Differential and Difference Equations, 29 (2001), 267. [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. 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, (1999). [18] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd edition,, Applied Mathematical Sciences, (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. doi: 10.1080/10236198.2010.549007. [20] S. Luzzatto and P. Pilarczyk, Finite resolution dynamics,, Found. Comput. Math., 11 (2011), 211. 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. 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., (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., (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. 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. 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. 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. 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, (2002). [29] R. Tarjan, Depth-first search and linear graph algorithms,, SIAM J. Comput., 1 (1972), 146. doi: 10.1137/0201010. [30] W. Tucker, A rigorous ODE solver and Smale's 14th problem,, Found. Comput. Math., 2 (2002), 53. doi: 10.1007/s002080010018. [31] W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations,, Princeton University Press, (2011). [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. doi: 10.1137/100795176. [33] J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay,, de Gruyter Series in Nonlinear Analysis and Applications, (2001). 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. 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). 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, (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. doi: 10.1080/10236198.2013.804916. [3] Y. Cao, Uniqueness of periodic solution for differential delay equations,, J. Differential Equations, 128 (1996), 46. 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. 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. 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. doi: 10.1007/s002110050240. [7] M. Dellnitz, A. Hohmann, O. Junge and M. Rumpf, Exploring invariant sets and invariant measures,, Chaos, 7 (1997), 221. 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. doi: 10.1080/10236190512331319334. [9] Z. Galias, Rigorous investigation of the Ikeda map by means of interval arithmetic,, Nonlinearity, 15 (2002), 1759. 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. doi: 10.1016/S0898-1221(04)90119-8. [11] S. Haykin, Neural Networks: A Comprehensive Foundation,, Prentice-Hall, (1999). [12] V. J. López, A counterexample on global attractivity for Clark's equation,, Proceedings of the Workshop Future Directions in Difference Equations, 69 (2011), 97. [13] V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications,, Mathematics and its Applications, (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., (1999), 1. [15] T. Krisztin, Unstable sets of periodic orbits and the global attractor for delayed feedback,, in Topics in Functional Differential and Difference Equations, 29 (2001), 267. [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. 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, (1999). [18] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd edition,, Applied Mathematical Sciences, (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. doi: 10.1080/10236198.2010.549007. [20] S. Luzzatto and P. Pilarczyk, Finite resolution dynamics,, Found. Comput. Math., 11 (2011), 211. 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. 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., (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., (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. 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. 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. 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. 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, (2002). [29] R. Tarjan, Depth-first search and linear graph algorithms,, SIAM J. Comput., 1 (1972), 146. doi: 10.1137/0201010. [30] W. Tucker, A rigorous ODE solver and Smale's 14th problem,, Found. Comput. Math., 2 (2002), 53. doi: 10.1007/s002080010018. [31] W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations,, Princeton University Press, (2011). [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. doi: 10.1137/100795176. [33] J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay,, de Gruyter Series in Nonlinear Analysis and Applications, (2001). 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. 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). 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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