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,, , (). 

show all references

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,, , (). 

[1]

Ricai Luo, Honglei Xu, Wu-Sheng Wang, Jie Sun, Wei Xu. A weak condition for global stability of delayed neural networks. Journal of Industrial and Management Optimization, 2016, 12 (2) : 505-514. doi: 10.3934/jimo.2016.12.505

[2]

Yunshyong Chow, Sophia Jang. Neimark-Sacker bifurcations in a host-parasitoid system with a host refuge. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1713-1728. doi: 10.3934/dcdsb.2016019

[3]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

[4]

Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks and Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295

[5]

Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457

[6]

Ozlem Faydasicok. Further stability analysis of neutral-type Cohen-Grossberg neural networks with multiple delays. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1245-1258. doi: 10.3934/dcdss.2020359

[7]

Maxime Breden, Laurent Desvillettes, Jean-Philippe Lessard. Rigorous numerics for nonlinear operators with tridiagonal dominant linear part. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4765-4789. doi: 10.3934/dcds.2015.35.4765

[8]

Jan Bouwe van den Berg, Ray Sheombarsing. Rigorous numerics for ODEs using Chebyshev series and domain decomposition. Journal of Computational Dynamics, 2021, 8 (3) : 353-401. doi: 10.3934/jcd.2021015

[9]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[10]

Leong-Kwan Li, Sally Shao. Convergence analysis of the weighted state space search algorithm for recurrent neural networks. Numerical Algebra, Control and Optimization, 2014, 4 (3) : 193-207. doi: 10.3934/naco.2014.4.193

[11]

Ivanka Stamova, Gani Stamov. On the stability of sets for reaction–diffusion Cohen–Grossberg delayed neural networks. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1429-1446. doi: 10.3934/dcdss.2020370

[12]

Sylvia Novo, Rafael Obaya, Ana M. Sanz. Exponential stability in non-autonomous delayed equations with applications to neural networks. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 517-536. doi: 10.3934/dcds.2007.18.517

[13]

Jui-Pin Tseng. Global asymptotic dynamics of a class of nonlinearly coupled neural networks with delays. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4693-4729. doi: 10.3934/dcds.2013.33.4693

[14]

Wenguo Shen. Unilateral global interval bifurcation for Kirchhoff type problems and its applications. Communications on Pure and Applied Analysis, 2018, 17 (1) : 21-37. doi: 10.3934/cpaa.2018002

[15]

Anna Belova. Rigorous enclosures of rotation numbers by interval methods. Journal of Computational Dynamics, 2016, 3 (1) : 81-91. doi: 10.3934/jcd.2016004

[16]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051

[17]

José A. Carrillo, Bertram Düring, Lisa Maria Kreusser, Carola-Bibiane Schönlieb. Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3985-4012. doi: 10.3934/dcds.2021025

[18]

Udhayakumar Kandasamy, Rakkiyappan Rajan. Hopf bifurcation of a fractional-order octonion-valued neural networks with time delays. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2537-2559. doi: 10.3934/dcdss.2020137

[19]

Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063

[20]

Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5567-5579. doi: 10.3934/dcdsb.2020367

 Impact Factor: 

Metrics

  • PDF downloads (139)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]