Hopf bifurcation and chaos analysis of Chen’s system with distributed delays. (English) Zbl 1080.34055

The paper studies bifurcations in delay differential equations (DDEs) with distributed delays. The local stability analysis is performed, using the Routh-Hurwitz citerion, only for the two cases of exponential integral kernels \(a\exp(-as)\) (called the weak kernel in the article) and \(a^2s\exp(-as)\) (called the strong kernel). Both of these special cases allow one to reduce the DDE to an ordinary differential equation (ODE). The analysis of the criticality of the Hopf bifurcation is performed for general kernels and then discussed more specifically for the two exponential kernels.


34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
Full Text: DOI


[1] Agiza, H. N.; Yessen, M. T., Synchronization of Rossler and Chen chaotic dynamical systems using active control, Phys Lett A, 278, 191-197 (2001) · Zbl 0972.37019
[2] Celikovsky, S.; Chen, G., On a generalized Lorenz canonical form of chaotic systems, Int J Bifurcat Chaos, 12, 1789-1812 (2002) · Zbl 1043.37023
[3] Chen, G.; Dong, X., From chaos to order: methodologies, perspectives and applications (1998), World Scientific: World Scientific Singapore · Zbl 0908.93005
[4] Chen, G.; Ueta, T., Yet another chaotic attractor, Int J Bifurcat Chaos, 9, 1465-1466 (1999) · Zbl 0962.37013
[5] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H., Theory and applications of Hopf bifurcation (1981), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0474.34002
[6] Krise, S.; Choudhury, S. R., Bifurcations and chaos in predator-prey model with delay and a laser-diode system with self-sustained pulsations, Chaos, Solitons & Fractals, 16, 59-77 (2003) · Zbl 1033.37048
[7] Li, C. P.; Chen, G., A note on Hopf bifurcation in Chen’s system, Int J Bifurcat Chaos, 13, 1609-1615 (2003) · Zbl 1074.34045
[8] Li, S. W.; Liao, X. F.; Li, C. G., Hopf bifurcation in a volterra prey-predator model with strong kernel, Chaos, Solitons & Fractals, 22, 713-722 (2004) · Zbl 1073.34086
[9] Li, T. C.; Chen, G., On stability and bifurcation of Chen’s system, Chaos, Solitons & Fractals, 19, 1269-1282 (2004) · Zbl 1069.34060
[10] Liao, X. F.; Wong, K. W.; Wu, Z. F., Bifurcation analysis on a two-neuron system with distributed delays, Physica D, 149, 123-141 (2001) · Zbl 1348.92035
[11] Liao, X. F.; Wong, K. W.; Wu, Z. F., Stability of bifurcating periodic solutions for van der Pol equation with continuous distributed delay, Appl Math Comput, 146, 313-334 (2001) · Zbl 1035.34083
[12] Liao, X. F.; Li, S. W.; Chen, G., Bifurcation analysis on a two-neuron system with distributed delays in the frequency domain, Neural Networks, 17, 545-561 (2004) · Zbl 1073.68715
[13] Lü, J. H.; Chen, G., A new chaotic attractor coined, Int J Bifurcat Chaos, 12, 659-661 (2002) · Zbl 1063.34510
[14] Lü, J. H.; Chen, G.; Celikovský, S., Bridge the gap between the Lorenz system and the Chen system, Int J Bifurcat Chaos, 12, 2917-2926 (2002) · Zbl 1043.37026
[15] Lü, J. H.; Zhou, T. S.; Chen, G.; Zhang, S. C., Local bifurcation of the Chen system, Int J Bifurcat Chaos, 12, 2257-2270 (2002) · Zbl 1047.34044
[16] Sparrow, C., The Lorenz equations: bifurcations, chaos, and strange attractors (1982), Springer-Verlag: Springer-Verlag NY · Zbl 0504.58001
[17] Ueta, T.; Chen, G., Bifurcation analysis of Chen’s attractor, Int J Bifurcat Chaos, 10, 1917-1931 (2000) · Zbl 1090.37531
[18] Vanecek, A.; Celikovský, S., Control systems: from linear analysis to synthesis of chaos (1996), Prentice-Hall: Prentice-Hall London · Zbl 0874.93006
[19] Yu, X.; Xia, Y., Detecting unstable periodic orbits in Chen’s chaotic attractor, Int J Bifurcat Chaos, 10, 1987-1991 (2000)
[20] Zhong, G. Q.; Tang, K. S., Circuitry implementation and synchronization of Chen’s attractor, Int J Bifurcat Chaos, 12, 1423-1427 (2002)
[21] Zhou, T. S.; Chen, G.; Tang, Y., Chen’s attractor exists, Int J Bifurcat Chaos, 14, 3167-3177 (2004) · Zbl 1129.37326
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