Hopf bifurcation analysis of a system of coupled delayed-differential equations

► A system of delay-differential equation involving a discrete time delay is studied. ► Hopf bifurcation is analyzed by choosing delay parameter as a bifurcation parameter. ► When delay parameter passes through some critical values, Hopf bifurcation occurs. ► The direction of bifurcation, the period...

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Bibliographic Details
Published in:Applied mathematics and computation 2013-02, Vol.219 (12), p.6605-6617
Main Authors: Çelik, C., Merdan, H.
Format: Article
Language:English
Subjects:
Bifurcations
Delay
Delay differential equation
Dynamical systems
Dynamics
Hopf bifurcation
Mathematical analysis
Mathematical models
Periodic solutions
Stability
Time delay
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Summary:► A system of delay-differential equation involving a discrete time delay is studied. ► Hopf bifurcation is analyzed by choosing delay parameter as a bifurcation parameter. ► When delay parameter passes through some critical values, Hopf bifurcation occurs. ► The direction of bifurcation, the period and the stability of solution are obtained. In this paper, we have considered a system of delay differential equations. The system without delayed arises in the Lengyel–Epstein model. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. Linear stability is investigated and existence of Hopf bifurcation is demonstrated via analyzing the associated characteristic equation. For the Hopf bifurcation analysis, the delay parameter is chosen as a bifurcation parameter. The stability of the bifurcating periodic solutions is determined by using the center manifold theorem and the normal form theory introduced by Hassard et al. (1981) [7]. Furthermore, the direction of the bifurcation, the stability and the period of periodic solutions are given. Finally, the theoretical results are supported by some numerical simulations.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2012.12.063