Dynamics of a two-dimensional delayed small-world network under delayed feedback control

International Journal of Bifurcation and Chaos

Volumn 16, Issue 11, 2006, Pages 3257-3273

  Xu, Xu ^a   Hu, Haiyan ^b   Wang, Huailei ^b  

^a JUIN UNIVERSITY   (China)

^b NANJING UNIVERSITY OF AERONAUTICS AND ASTRONAUTICS   (China)

Author keywords

Center manifold theorem; Normal form theory; Small world network; Stability switches; Time delay

Indexed keywords

FEEDBACK CONTROL; HOPF BIFURCATION; STABILITY CRITERIA; THEOREM PROVING;

CENTER MANIFOLD THEOREM; NORMAL FORM THEORY; SMALL-WORLD NETWORK; STABILITY SWITCHES;

TIME DELAY;

EID: 33847682304     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S021812740601677X     Document Type: Article

References (28)

1. Atay, F. M., Jost, J. & Wende, A. [2004] "Delays, connection topology, and synchronization of coupled chaotic maps," Phys. Rev. Lett. 92, 144101-144104.
2. Belykh, I. V., Belykh, V. N. & Hasler, M. [2004] "Blinking model and synchronization in small-world networks with a time-varying coupling," Physica D 195, 188-206.
3. Campbel, S. A. & Bélair, J. [1999] "Resonant codimension two bifurcation in the harmonic oscillator with delayed forcing," Canadian Appl. Math. Quart. 7, 218-238.
4. Collins, J. J. & Chow, C. C. [1998] "It's a small-world," Nature (London) 393, 409-410.
5. Ermentrout, B. [1997] XPPAUT 3.0 - The Differential Equations Tool (University of Pittsburgh, Pittsburgh).
6. Hale, J. K. [1997] Theory of Functional Differential Equations (Springer-Verlag, NY).
7. Hassard, B. D., Kazarinoff, N. D. & Wan, Y. H. [1981] Theory and Applications of Hopf Bifurcation (Cambridge University Press, Cambridge).
8. Herrero, C. P. [2002] "Ising model in small-world networks," Phys. Rev. E 65, 066110-066116.
9. Hu, H. Y. & Wang, Z. H. [2002] Dynamics of Controlled Mechanical Systems with Delayed Feedback (Springer-Verlag, Heidelberg).
10. Kulkarni, R. V., Almaas, E. & Stroud, D. [2000] "Stroud, exact results and scaling properties of small-world networks," Phys. Rev. E 61, 4268-4271.
11. Li, C. & Chen, G. [2004a] "Phase synchronization in small-world networks of chaotic oscillators," Physica A 341, 73-79.
12. Li, C. & Chen, G. [2004b] "Local stability and Hopf bifurcation in small-world delayed networks," Chaos Solit. Fract. 20, 353-361.
13. Li, C. G., Xu, H. B., Liao, X. F. & Yu, J. B. [2004] "Synchronization in small-world oscillator networks with coupling delays," Physica A 335, 359-364.
14. Moukarzel, C. F. [1999] "Spreading and shortest paths in systems with sparse long-range connections," Phys. Rev. E 60, 6263-6266.
15. Newman, M. E. J. & Watts, D. J. [1999] "Scaling and percolation in the small-world network model," Phys. Rev. E 60, 7332-7342.
16. Newman, M. E. J., Morre, C. & Watts, D. J. [2000] "Mean-field solution of the small-world network model," Phys. Rev. Lett. 84, 3201-3204.
17. Newman, M. E. J., Jensen, I. & Ziff, R. M. [2002] "Percolation and epidemics in a two-dimensional small-world," Phys. Rev. E 65, 021904.
18. Pandit, S. A. & Amritkar, R. E. [1999] "Characterization and control of small-world networks," Phys. Rev. E 60, 1119-1122.
19. Pyragas, K. [1992] "Continuous control of chaos by self-controlling feedback," Phys. Lett. A 170, 421-428.
20. Shefi, O., Golding, I., Segev, R., Ben-Jacob, E. & Ayali, A. [2002] "Morphological characterization of in vitro neuronal networks," Phys. Rev. E 66, 021905.
21. Wang, X. F. & Chen, G. [2003] "Complex networks: Small-world, scale-free and beyond," IEEE Circuits Syst. Mag. First Quarter, 6-20.
22. Wang, X. F. & Chen, G. [2002] "Synchronization in small-world dynamical networks," Int. J. Bifurcation and Chaos 12, 187-192.
23. Wang, Z. H. & Hu, H. Y. [2000] "Stability switches of time-delayed dynamic systems with unknown parameters," J. Sound Vibr. 233, 215-233.
24. Watts, D. J. & Strogatz, S. H. [1998] "Collective dynamics of 'small-world' networks," Nature 393, 440-442.
25. Xu, J. & Chung, K. W. [2003] "Effects of time delayed position feedback on a van der Pol-Duffing oscillator," Physica D 180, 17-39.
26. Yang, X. S. [2001] "Chaos in small-world networks," Phys. Rev. E 63, 046206.
27. Yang, X. H. [2002] "Fractals in small-world networks with time-delay," Chaos Solit. Fract. 13, 215-219.
28. Zekri, N., Porterie, B., Clerc, J. P. & Loraud, J. C. [2005] "Propagation in a two-dimensional weighted local small-world network," Phys. Rev. E 71, 046121-046125.