On periodic solutions of a two-neuron network system with sigmoidal activation functions

International Journal of Bifurcation and Chaos

Volumn 16, Issue 5, 2006, Pages 1405-1417

  Hsu, Cheng Hsiung ^a   Yang, Suh Yuh ^a   Yang, Ting Hui ^b   Yang, Tzi Sheng ^c  

^a NATIONAL CENTRAL UNIVERSITY   (Taiwan)

^b TAMKANG UNIVERSITY   (Taiwan)

^c TUNGHAI UNIVERSITY   (Taiwan)

Author keywords

Contraction mapping theorem; Dulac's criterion; Liapunov functions; Neural networks; Periodic solutions; Poincar Bendixson theorem

Indexed keywords

CONFORMAL MAPPING; FEEDBACK; FUNCTIONS; LYAPUNOV METHODS; NUMERICAL METHODS; SYSTEM STABILITY; THEOREM PROVING; TIME VARYING SYSTEMS;

CONTRACTION MAPPING THEOREM; DULAC'S CRITERION; PERIODIC SOLUTIONS; POINCARÉ-BENDIXSON THEOREM;

NEURAL NETWORKS;

EID: 33747460007     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127406015386     Document Type: Article

References (23)

1. Bélair, J., Campbell, S. A. & Van den Driessche, P. [1996] "Frustration, stability, and delay-induced oscillations in a neural network model," SIAM J. Appl. Math. 56, 245-255.
2. Cao, J. & Li, Q. [2000] "On the exponential stability and periodic solutions of delayed cellular neural networks," J. Math. Anal. Appl. 252, 50-64.
3. Chua, L. O. [1998] CAW: A Paradigm for Complexity, World Scientific Series on Nonlinear Science, Series A, Vol. 31 (World Scientific, Singapore).
4. Chua, L. O. & Yang, L. [1998a] "Cellular neural networks: Theory," IEEE Trans. Circuits Syst. 35, 1257-1272.
5. Chua, L. O. & Yang, L. [1998b] "Cellular neural networks: Applications," IEEE Trans. Circuits Syst. 35, 1273-1290.
6. Civalleri, P. P. & Gilli, M. [1993] "On the dynamic behavior of two-cell cellular neural networks," Int. J. Circuits Th. Appl. 21, 451-471.
7. Gopalsamy, K. & Leung, I. [1996] "Delay induced periodicity in a neural netlet of excitation and inhibition," Physica D 89, 395-426.
8. Hale, J. & Koçak, H. [1991] Dynamics and Bifurcations, Texts in Applied Mathematics, Vol. 3 (Springer-Verlag, NY).
9. Hopfield, J. J. [1984] "Neurons with graded response have collective computational properties like two-state neurons," Proc. Natl. Acad. Sci. U.S.A. 81, 3088-3092.
10. Hu, T. & Lin, Z. [2000] "A complete stability analysis of planar systems under saturation," IEEE Trans. Circuits Syst. 47, 498-512.
11. Lin, S.-S. & Shih, C.-W. [1999] "Complete stability for standard cellular neural networks," Int. J. Bifurcation and Chaos 9, 909-918.
12. Lin, S.-S. & Lin, Y.-H. [2004] "Uniqueness of exterior limit cycle in two-cell cellular neural network," preprint.
13. Lin, S.-S., Lin, W.-W. & Yang, T.-H. [2004] "Bifurcations and chaos in two-cells cellular neural networks with periodic inputs," Int. J. Bifurcation and Chaos 14, 3179-3204.
14. Marcus, C. M. & Westervelt, R. M. [1989] "Stability of analog neural networks with delay," Phys. Rev. A 39, 347-359.
15. Marcus, C. M., Waugh, F. R. & Westervelt, R. M. [1991] "Nonlinear dynamics and stability of analog neural networks," Physica D 51, 234-247.
16. Mastsuoka, K. [1991] "Stability conditions for nonlinear continuous neural networks with asymmetric connection weight," Neural Networks 5, 495-500.
17. Olien, L. & Bélair, J. [1997] "Bifurcations, stability, and monotonicity properties of a delayed neural network model," Physica D 102, 349-363.
18. Perko, L. [1991] Differential Equations and Dynamical Systems (Springer-Verlag, NY).
19. Shayer, L. P. & Campbell, S. A. [2000] "Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays," SIAM J. Appl. Math. 61, 673-700.
20. Shih, C.-W. [2001] "Complete stability for a class of cellular neural networks," Int. J. Bifurcation and Chaos 11, 169-177.
21. Wang, X. & Blum, E. K. "Discrete-time versus continuous time models of neural networks," J. Comp. Syst. Sci. 45, 1-19.
22. Ye, H., Michel, A. N. & Wang, K. [1994] "Global stability and local stability of Hopfield neural networks with delays," Phys. Rev. E 50, 4206-4213.
23. Zou, F. & Nossek, J. [1991] "Stability of cellular neural networks with opposite-sign templates," IEEE Trans. Circuits Syst. 38, 675-677.