Stability and Neimark-Sacker Bifurcation Analysis in a Genetic Network with Delay

Feng Liu; Xiang Yin; Zhe Zhang; Fenglan Sun

doi:10.20965/jaciii.2017.p0278

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JACIII Vol.21 No.2 pp. 278-283

doi: 10.20965/jaciii.2017.p0278

(2017)

Paper:

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Stability and Neimark-Sacker Bifurcation Analysis in a Genetic Network with Delay

Feng Liu^,†, Xiang Yin^, Zhe Zhang^*, and Fenglan Sun^**

^*School of Automation, China University of Geosciences
No.388 Lumo Road, Wuhan 430074, China

^**Research Center of System Theory and Application, Chongqing University of Posts and Telecommunications
No.2 Chongwen Road, Nanan District, Chongqing 400065, China

^†Corresponding author

Received:

July 7, 2016

Accepted:

November 10, 2016

Online released:

March 15, 2017

Published:

March 20, 2017

Keywords:

stability, genetic network, Neimark-Sacker bifurcation, delay

Abstract

This paper investigates a genetic model with delay. The stability, direction, and bifurcation periodic solution is derived by using the center manifold theorem and normal form theory. Numerical simulations illustrate the theoretical results.

Cite this article as:

F. Liu, X. Yin, Z. Zhang, and F. Sun, “Stability and Neimark-Sacker Bifurcation Analysis in a Genetic Network with Delay,” J. Adv. Comput. Intell. Intell. Inform., Vol.21 No.2, pp. 278-283, 2017.

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References

[1] J. K. Hale and S. V. Lunel, “Introduction to Functional Differential Equations,” Springer-Verlag, 1993.
[2] K. Gu, V. Kharitonov, and J. Chen, “Stability of Time-Delay Systems,” Birkhäuser Basel , 2003.
[3] S. Haykin, “Neural Networks,” Prentice Hall, 1999.
[4] S. Niculescu, “Delay Effects on Stability: A Robust Control Approach,” Springer-Verlag, 2001.
[5] L. Chen and K. Aihara, “Stability of genetic regulatory networks with time delay,” IEEE Trans. Circuits & Systems I, Vol.49, No.5, pp. 602-608, 2002.
[6] M. Xiao and J. Cao, “Genetic oscillation deduced from Hopf bifurcation in a genetic regulatory network with delays,” Mathematical biosciences, Vol.215, No.1, pp. 55-63, 2008.
[7] F. Wu, “Global and robust stability analysis of genetic regulatory networks with time-varying delays and parameter uncertainties,” IEEE Trans. Biomedical Circuits and Systems, Vol.5, No.4, pp. 391-398, 2011.
[8] W. Wang and S. Zhong, “Stochastic stability analysis of uncertain genetic regulatory networks with mixed time-varying delays,” Neurocomputing, Vol.82, pp. 143-156, 2012.
[9] P. Smolen, D. A. Baxter, and J. H. Byrne, “Modeling transcriptional control in gene networks-methods, recent results, and future,” Bull. Math. Biol., Vol.62, pp. 247-292, 2007.
[10] C. Zhang, M. Liu, and B. Zheng, “Hopf bifurcation in numerical approximation of a class delay differential equations,” Appl. Math. Comput., Vol.146, pp. 335-349, 2003.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] J. K. Hale and S. V. Lunel, “Introduction to Functional Differential Equations,” Springer-Verlag, 1993.

[2] [2] K. Gu, V. Kharitonov, and J. Chen, “Stability of Time-Delay Systems,” Birkhäuser Basel , 2003.

[3] [3] S. Haykin, “Neural Networks,” Prentice Hall, 1999.

[4] [4] S. Niculescu, “Delay Effects on Stability: A Robust Control Approach,” Springer-Verlag, 2001.

[5] [5] L. Chen and K. Aihara, “Stability of genetic regulatory networks with time delay,” IEEE Trans. Circuits & Systems I, Vol.49, No.5, pp. 602-608, 2002.

[6] [6] M. Xiao and J. Cao, “Genetic oscillation deduced from Hopf bifurcation in a genetic regulatory network with delays,” Mathematical biosciences, Vol.215, No.1, pp. 55-63, 2008.

[7] [7] F. Wu, “Global and robust stability analysis of genetic regulatory networks with time-varying delays and parameter uncertainties,” IEEE Trans. Biomedical Circuits and Systems, Vol.5, No.4, pp. 391-398, 2011.

[8] [8] W. Wang and S. Zhong, “Stochastic stability analysis of uncertain genetic regulatory networks with mixed time-varying delays,” Neurocomputing, Vol.82, pp. 143-156, 2012.

[9] [9] P. Smolen, D. A. Baxter, and J. H. Byrne, “Modeling transcriptional control in gene networks-methods, recent results, and future,” Bull. Math. Biol., Vol.62, pp. 247-292, 2007.

[10] [10] C. Zhang, M. Liu, and B. Zheng, “Hopf bifurcation in numerical approximation of a class delay differential equations,” Appl. Math. Comput., Vol.146, pp. 335-349, 2003.

Stability and Neimark-Sacker Bifurcation Analysis in a Genetic Network with Delay

Feng Liu*,†, Xiang Yin*, Zhe Zhang*, and Fenglan Sun**

Feng Liu^,†, Xiang Yin^, Zhe Zhang^*, and Fenglan Sun^**