Delay Induced Hopf Bifurcation of Small-World Networks

Abstract

In this paper, the stability and the Hopf bifurcation of small-world networks with time delay are studied. By analyzing the change of delay, we obtain several sufficient conditions on stable and unstable properties. When the delay passes a critical value, a Hopf bifurcation may appear. Furthermore, the direction and the stability of bifurcating periodic solutions are investigated by the normal form theory and the center manifold reduction. At last, by numerical simulations, we further illustrate the effectiveness of theorems in this paper.

Keywords

Small-world networks / Time delay / Hopf bifurcation / 34K15

References

[1] Watts Nature, 1998, 393: 440

[2] Nnewman Physical Review E, 1999, 60: 7332

[3] Yang X. S.. Chaos in small-world networks. Physical Review E, 2001, 63: 046206

[4] Wang Int. J. Bifur. Chaos, 2002, 12: 189

[5] Li Physical Review E, 2003, 68: 052901

[6] Hassard, B. D., Kazarinoff, N. D. andWan, Y. H., Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981

[7] Wei Physica D, 1999, 130: 255

[8] Liao Physica D, 2001, 149: 123

[9] Li Chaos, Solitons and Fractals, 2004, 20: 353

[10] Meng Chaos, Solitons and Fractals, 2004, 21: 729

[11] Wei Acta Mathematica Sinica, 2002, 45: 93

[12] Hale, J. K. and Lunel, S. V., Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993

[13] Liu Chin. Ann. Math., 2005, 26B: 253