Synchronization and desynchronization in a delayed discrete neural network

International Journal of Bifurcation and Chaos

Volumn 17, Issue 3, 2007, Pages 791-803

  Peng, Mingshu ^a   Yuan, Yuan ^b  

^a BEIJING JIAOTONG UNIVERSITY   (China)

^b MEMORIAL UNIVERSITY OF NEWFOUNDLAND   (Canada)

Author keywords

Delay feedback; Discrete neural network; Multiple periodic patterns; Pitchfork flip Neimark Sacker bifurcation

Indexed keywords

ASYMPTOTIC ANALYSIS; BIFURCATION (MATHEMATICS); FEEDBACK; HARMONIC ANALYSIS; SYNCHRONIZATION; TIME DELAY;

DELAYED DISCRETE NEURAL NETWORKS; EXCITORY INTERACTIONS; PERIODIC PATTERNS;

NEURAL NETWORKS;

EID: 33750612172     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127407017550     Document Type: Article

References (34)

1. Baldi, P. & Atiya, A. F. [1994] "How delays affect neural dynamics and learning," IEEE Trans. Neural Networks 5, 612-621.
2. Carr, J. [1981] Applications of Center Manifold Theory (Springer-Verlag, NY).
3. Clark, C. W. [1976] "A delay-recruitment model of population dynamics with an application to baleen whale populations," J. Math. Biol. 5, 612-621.
4. Fick, E., Fick, M. & Hausmann, G. [1991] "Logistic equation with memory," Phys. Rev. A 44, 2469-2473.
5. Foss, J., Longtin, A., Mensour, B. & Milton, J. [1996] "Multistability and delayed recurrent loops," Phys. Rev. Lett. 76, 708-711.
6. Foss, J., Moss, F. & Milton, J. [1997] "Noise, multistability and delayed recurrent loops," Phys. Rev. E 55, 4536-4543.
7. Foss, J., Moss, F. & Milton, J. [2000] "Multistability in recurrent neural loops arising from delay," J. Neuronphysiol. 84, 975-985.
8. Hopfield, J. J. [1982] "Neural networks and physical systems with emergent collective computational abilities," Proc. Natl. Acad. Sci. 79, 2554-2558.
9. Hopfield, J. J. [1984] "Neurons with graded response have collective computational properties like those of two-state neurons," Proc. Natl. Acad. Sci. 81, 3088-3092.
10. Iooss, G. [1979] Bifurcation of Maps and Applications (North-Holland, Amsterdam).
11. Koto, T. [1999] "Neimark-Sacker bifurcations in the Euler method for a delay differential equation," BIT 39, 110-115.
12. Koto, T. [2001] "Periodic orbits in the Euler methods for a class of delay differential equations," Comput. Math. Appl. 42, 1597-1608.
13. n-k = 0," J. Math. Anal. Appl. 188, 719-731.
14. Kuznetsov, Y. A. [1995] Elements of Applied Bifurcation Theory (Springer-Verlag, NY).
15. Levin, S. A. & May, R. M. [1976] "A note on difference-delay equations," Theoret. Popul. Biol. 9, 178-187.
16. Masoller, C. & Zanette, D. H. [2001] "Anticipated synchronization in coupled chaotic maps with delays," Physica A 300, 359-366.
17. Milton, J. [1996] Dynamics of Small Neural Populations CRM Monograph Series, Vol. 7. (American Mathematical Society, Providence, RI).
18. Morita, M. [1993] "Associate memory with nonmonotone dynamics," Neural Networks 6, 115-126.
19. Peng, M. [2004a] "Bifurcation and chaotic behavior in the Euler method for a Uçar prototype delay model," Chaos Solit. Fract. 22, 483-493.
20. Peng, M. [2004b] "Bifurcation and chaotic behavior in the Euler method for a Kaplan-Yorke prototype delay model," Chaos Solit. Pract. 20, 489-496.
21. Peng, M., Bai, E. W. & Lonngren, K. E. [2004] "On the synchronization of delay discrete models," Chaos Solit. Fract. 22, 573-576.
22. Peng, M. [2005a] "Multiple bifurcations and periodic 'bubbling' in a delay population model," Chaos Solit. Fract. 25, 1123-1130.
23. Peng, M. [2005b] "Rich dynamics of discrete delay ecological models," Chaos Solit. Fract. 24, 1279-1285.
24. Peng, M. [2005c] "Symmetry breaking, bifurcations, periodicity and chaos in the Euler method for a class of delay differential equations," Chaos Solit. Pract. 24, 1287-1297.
25. Stewart, I. [2003] Galois Theory, 3rd edition (Chapman & Hall/CRC Mathematics, London)
26. Wiggins, S. [2003] Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition (Springer-Verlag, Berlin).
27. Wu, J., Faria T. & Huang, Y. S. [1999] "Synchronization and stable phase-locking in a network of neurons with memory," Math. Comput. Model. 30, 117-138.
28. Wu, J. [2001] Introduction to Neural Dynamics and Signal Transmission Delay (De-Gruyter, Berlin).
29. Wu, J. & Zhang, R. [2004] "A simple delayed network with large capacity for associate memory," Discr. Cont. Dyn. Syst. B 4, 853-865.
30. Wu, J., Zhang, R. & Zou, X. [2004] "Multiple periodic patterns via discrete neural nets with delayed feedback loops," Int. J. Bifurcation and Chaos 14, 2915-2923.
31. Yuan, Y. & Campbell, S. A. [2004] "Stability and synchronization of a ring of identical cells with delayed coupling," J. Dyn. Diff. Eqs. 16, 709-744.
32. Yuan, Y. & Wei, J. [2006] "Multiple bifurcation analysis in a neural network model with delay," Int. J. Bifurcation and Chaos 16, 2903-2913.
33. Zou, Z. & Wu, J. [2003] "Attractive periodic orbits in nonlinear discrete-time networks with delayed feedback," J. Diff. Eqs. Appl. 8, 467-483.
34. Zou, Z. & Wu, J. [2003] "Attractive periodic orbits for discrete monotone dynamical systems arising from delayed neural works," Dyn. Cont. Discr. Impul. Syst. 10, 95-106.