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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 |
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). Google Scholar |
[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). Google Scholar |
[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,, , (). Google Scholar |
[38] |
Computer Assisted Proofs in Dynamics group, CAPD Library,, , (). Google Scholar |
[39] |
National Information Infrastructure Development Institute, NIIF,, , (). Google Scholar |
show all references
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). Google Scholar |
[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). Google Scholar |
[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,, , (). 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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