Complex dynamics in discrete delayed models with D4 symmetry

Chaos, Solitons and Fractals

Volumn 37, Issue 2, 2008, Pages 393-408

  Peng, Mingshu ^a   Yuan, Yuan ^b  

^a BEIJING JIAOTONG UNIVERSITY   (China)

^b MEMORIAL UNIVERSITY OF NEWFOUNDLAND   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

BIFURCATION (MATHEMATICS); FEEDBACK CONTROL; MATHEMATICAL MODELS; NONLINEAR SYSTEMS; RESONATORS; VECTORS;

DISCRETE DELAYED MODELS; NEIMARK-SACKER BIFURCATIONS; OPTICAL MODELS; PATTERN FORMATION; VECTOR SPACES;

TIME DELAY;

EID: 40749160948     PISSN: 09600779     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.chaos.2006.08.048     Document Type: Article

References (32)

1. Buldú J.M., García-Ojalvo J., and Torrent M.C. Delay-induced resonances in an optical system with feedback. Phys Rev E 69 (2004) 046207 1-4
2. Campbell S.A., Yuan Y., and Bungay S. Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling. Nonlinearity 18 (2005) 2827-2846
3. Destexhe A. Oscillations, complex spatiotemporal behavior, and information transport in networks of excitory and inhibitory neurons. Phys Rev E 50 2 (1994) 1594-1606
4. Foss J., Longtin A., Mensour B., and Milton J. Multistability and delayed recurrent loops. Phys Rev Lett 76 (1996) 708-711
5. Golubitsky M., Stewart I., and Schaeffer D.G. Singularities and groups in bifurcation theory, vol. II. Applied mathematical sciences 69 (1988), Springer-Verlag, New York
6. Gray C.M., König P., Engel A.K., and Singer W. Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature (London) 338 (1989) 334-336
7. Guo S. Spatio-temporal patterns of nonlinear oscillations in an excitatory ring network with delay. Nonlinearity 18 5 (2005) 2391-2407
8. Guo S., and Huang L. Hopf bifurcating periodic orbits in a ring of neurons with delays. Physica D 183 (2003) 19-44
9. Huang L., and Wu J. Nonlinear waves in networks of neurons with delayed feedback: pattern formation and continuation. SIAM J Math Anal 34 4 (2003) 836-860
10. Ikeda K., and Akimoto O. Instability leading to periodic and chaotic self-pulsations in a bistable optical cavity. Phys Rev Lett 48 9 (1982) 617-620
11. Ikeda K., and Matsumoto K. Study of a high-dimensional chaotic attractor. J Stat Phys 44 5-6 (1986) 955-983
12. Kim S., Park S.H., and Ryu C.S. Multistability in coupled oscillator systems with time delay. Phys Rev Lett 79 (1997) 2911-2914
13. k = 0. J Math Anal Appl 188 (1994) 719-731
14. Kuznetsov Y.A. Elements of applied bifurcation theory (1995), Springer-Verlag, New York
15. Lamb J., and Roberts J. Time-reversible and equivariant pitchfork bifurcation: a survey. Physica D 112 1-2 (1998) 1-39
16. Lim C.C., and McComb I. Time-reversal symmetry in dynamical systems. Physica D 112 (1998) 117-121
17. Masoller C., and Zanette D.H. Anticipated synchronization in coupled chaotic maps with delays. Physica A (2001) 359-366
18. Niebur E., Schuster H.G., and Kammen D.M. Collective frequencies and metastability in networks of limit-cycle oscillators with time delay. Phys Rev Lett 67 (1991) 2753-2756
19. Peng M. Effective approaches to explore rich dynamics of delay differential equations. Chaos, Solitons & Fractals 25 (2005) 1131-1140
20. Peng M. Symmetry breaking, bifurcations periodicity and chaos in the Euler method for a class of delay differential equations. Chaos, Solitons & Fractals 24 (2005) 1287-1297
21. Peng M. Multiple bifurcations and periodic 'bubbling' in a delay population model. Chaos, Solitons & Fractals 25 (2005) 1123-1130
22. Peng M. Rich dynamics of discrete delay ecological models. Chaos, Solitons & Fractals 24 (2005) 1279-1285
23. Peng M. Bifurcation and chaotic behavior in the Euler method for a Kaplan-Yorke prototype delay model. Chaos, Solitons & Fractals 20 (2004) 489-496
24. Peng M., Bai E.W., and Lonngren K.E. On the synchronization of delay discrete models. Chaos, Solitons & Fractals 22 (2004) 573-576
25. Peng M., and Yuan Y. Synchronization and desynchronization in a delayed discrete neural network. Int J Bifurcations Chaos Appl Sci Eng 17 3 (2007)
26. Peng M, Yuan Y. Stability, symmetry-breaking bifurcation and chaos in discrete delayed models, submitted for publication.
27. Sompolinsky H., Golomb D., and Kleinfeld D. Cooperative dynamics in visual processing. Phys Rev A 43 (1991) 6990-7011
28. Stone L. Period-doubling reversals and chaos in simple ecological models. Nature 363 (1993) 411-441
29. Ermentrout D.H.T.; G.B., and Yew A.C. Propagating activity patterns in thalamic neuronal networks. SIAM J Appl Math 61 5 (2001) 1578-1604
30. Wu J. Symmetric functional-differential equations and neural networks with memory. Trans Amer Math Soc 350 (1998) 4799-4838
31. Wu J., and Zhang R. Simple delayed network with large capacity for associate memory. Discr Cont Dyn Syst B 4 (2004) 853-865
32. Wu J., Zhang R., and Zou X. Multiple periodic patterns via discrete neural nets with delayed feedback loops. Int J Bifurction Chaos Appl Sci Eng 14 (2004) 2915-2923