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The problem of Hopf bifurcation control for a predator-prey system with three delays is considered. A new hybrid strategy is proposed to control the Hopf bifurcation, in which the state feedback and parameter perturbation are used to delay the onset of an inherent bifurcation or make the bifurcation disappear. The stability and the existence of bifurcation are researched. In particular, the formulae determining the direction of the bifurcations and the stability of the bifurcating periodic solutions are derived by using the normal form theory and center manifold theorem. Finally, numerical simulation results confirm that the new hybrid controller is efficient in controlling Hopf bifurcation.
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Keywords:
- delay /
- predator-prey system /
- Hopf bifurcation /
- hybrid control
[1] May R M 1973 Ecology 54 315
[2] Sarker R, Petrovskii S, Biswas M, Gupta A, Chattopadhyay J 2006 Ecol Model 193 589
[3] Luo X S, Chen G R, Wang B H 2003 Chaos Solit. Fract. 18 775
[4] Liu Z R, Chung K W 2005 Int. J. Bifurcation Chaos 15 3895
[5] Sun B D 2005 Principle of Automatic Control (Beking:China Machine Press) (in Chinese)[孙炳达 2005 自动控制原理 (北京: 机械工业出版社)]
[6] Faria T 2001 Math. Anal. Appl. 254 433
[7] Hassard B D, Azarinoff N D K, Wan Y H 1981 Theory and Applications of Hopf Bifurcation (Cambridge University Press, Cambridge)
[8] Freedman H I, Rao V S H 1986 SIAM J. Appl. Math. 46 552
[9] Hale J K 1977 Theory of Functional Differential Equations (NewYork: Springer)
[10] He X 1996 Math. Anal. Appl. 198 355
[11] Mao Z S, Zhao H Y 2007 Phys. Lett. A 364 38
[12] Zhao H Y, Chen L, Mao Z S 2009 Nonlinear Analysis: Real World Applications 9 663
[13] Song Y L, Wei J J 2005 Math. Anal. Appl. 301 1
[14] Yan X P, Li W T 2006 Appl. Math. Comput. 177 427
[15] Mao Z S, Zhao H Y, Wang X F 2007 Physica D 234 11
[16] Yan X P, Zhang C H 2008 Nonlinear Anal. 9 114
[17] Ruan S G, Wei J J 2003 Dynamics Continuous, Discrete Impulsive Systems Ser. A: Math. Anal. 10 863
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[1] May R M 1973 Ecology 54 315
[2] Sarker R, Petrovskii S, Biswas M, Gupta A, Chattopadhyay J 2006 Ecol Model 193 589
[3] Luo X S, Chen G R, Wang B H 2003 Chaos Solit. Fract. 18 775
[4] Liu Z R, Chung K W 2005 Int. J. Bifurcation Chaos 15 3895
[5] Sun B D 2005 Principle of Automatic Control (Beking:China Machine Press) (in Chinese)[孙炳达 2005 自动控制原理 (北京: 机械工业出版社)]
[6] Faria T 2001 Math. Anal. Appl. 254 433
[7] Hassard B D, Azarinoff N D K, Wan Y H 1981 Theory and Applications of Hopf Bifurcation (Cambridge University Press, Cambridge)
[8] Freedman H I, Rao V S H 1986 SIAM J. Appl. Math. 46 552
[9] Hale J K 1977 Theory of Functional Differential Equations (NewYork: Springer)
[10] He X 1996 Math. Anal. Appl. 198 355
[11] Mao Z S, Zhao H Y 2007 Phys. Lett. A 364 38
[12] Zhao H Y, Chen L, Mao Z S 2009 Nonlinear Analysis: Real World Applications 9 663
[13] Song Y L, Wei J J 2005 Math. Anal. Appl. 301 1
[14] Yan X P, Li W T 2006 Appl. Math. Comput. 177 427
[15] Mao Z S, Zhao H Y, Wang X F 2007 Physica D 234 11
[16] Yan X P, Zhang C H 2008 Nonlinear Anal. 9 114
[17] Ruan S G, Wei J J 2003 Dynamics Continuous, Discrete Impulsive Systems Ser. A: Math. Anal. 10 863
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