Bifurcation analysis for Chen's system with delayed feedback and its application to control of chaos☆

Song, Yongli; Wei, Junjie

doi:10.1016/j.chaos.2003.12.075

Cited by 176 publications

(63 citation statements)

References 11 publications

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“…The proof of Lemma 2.9 is similar to that in the proof of Lemma 8 in [19], here we omit it. Then, from the above discussion and Lemmas 2.6-2.9, we have the following theorem.…”

Section: And D(reλ(τ (J)mentioning

confidence: 80%

“…Due to the hyperbolicity of E 0 , it is not part of any cycle chain in R 3 + . Thus, every bounded forward orbits of Equation (19) in R 3 + converges towards the unique endemic equilibrium E * . Therefore, E * is global asymptotically stable.…”

Section: Theorem 25mentioning

confidence: 99%

“…Then system (19) of ordinary differential equations has nonconstant periodic solution. System (19) have the same equilibria to system (3), i.e. E 0 ( /d, 0, 0) and a positive equilibrium E * (S * , I * , R * ).…”

Section: Proofmentioning

confidence: 99%

“…Noting that the variable R only occurs in the third equation of system (19), from Lemma 2.4, it is enough to investigate the subsystem…”

Section: Theorem 25mentioning

confidence: 99%

“…In what follows, using time delay as the bifurcation parameter, we investigate the Hopf bifurcation for system (3) and the stability of E * by using the method in [16,19].…”

Section: Theorem 25mentioning

confidence: 99%

See 4 more Smart Citations

Global Hopf bifurcation analysis of an susceptible-infective-removed epidemic model incorporating media coverage with time delay

Zhao

2016

Journal of Biological Dynamics

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“…The proof of Lemma 2.9 is similar to that in the proof of Lemma 8 in [19], here we omit it. Then, from the above discussion and Lemmas 2.6-2.9, we have the following theorem.…”

Section: And D(reλ(τ (J)mentioning

confidence: 80%

Section: Theorem 25mentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

“…Noting that the variable R only occurs in the third equation of system (19), from Lemma 2.4, it is enough to investigate the subsystem…”

Section: Theorem 25mentioning

confidence: 99%

“…In what follows, using time delay as the bifurcation parameter, we investigate the Hopf bifurcation for system (3) and the stability of E * by using the method in [16,19].…”

Section: Theorem 25mentioning

confidence: 99%

See 3 more Smart Citations

Global Hopf bifurcation analysis of an susceptible-infective-removed epidemic model incorporating media coverage with time delay

Zhao

2016

Journal of Biological Dynamics

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Stability and Hopf bifurcation analysis of novel hyperchaotic system with delayed feedback control

Mani

Balasubramaniam

2015

Complexity

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In this article, a novel four dimensional autonomous nonlinear systezm called hyperchaotic Rikitake system is proposed. Basic properties of the new system are investigated and the complex dynamical behaviors, such as time series, bifurcation diagram, and Lyapunov exponents are analyzed by dynamic analysis approaches. To control the new hyperchaotic system, the delayed feedback control is introduced. Regarding the time delay as a bifurcation parameter, stability and bifurcations with respect to time delay are investigated. Conditions assuring the existence of Hopf bifurcation and the distribution of roots to the associated characteristic equation are investigated by utilizing the polynomial theorem. Besides, the Hopf bifurcation is proved to occur when the bifurcation parameter (time delay) crosses through derived critical value. Finally, numerical simulations are provided to prove the consistence with the derived theoretical results. © 2015 Wiley Periodicals, Inc. Complexity 21: 180–193, 2016

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Hopf bifurcation and global stability of density‐dependent model with discrete delays involving Beddington‐DeAngelis functional response

Singh

Parwaliya

Kumar

2021

Math Methods in App Sciences

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A prey‐predator model with two discrete delays incorporating Beddington‐DeAngelis functional response is investigated. The positivity and boundedness of the solution of the delayed system have been discussed. Further, permanence of the system has been established under sufficient conditions which proves long‐term survival of prey and predator population. The occurrence of periodic solutions in the system is confirmed via phenomenon of Hopf bifurcation with respect to both discrete delays. The properties of periodic solutions are determined by using central manifold and normal form theory. The supercritical Hopf bifurcation occurs with respect to one delay parameter. The global stability of the system has been established for different combinations of delays. The numerical computation has also been performed to verify analytical results. The presence of delays in the system causes a wide range of complex dynamics, viz., limit cycles, quasi‐periodicity, and chaos.

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Bifurcation analysis for Chen's system with delayed feedback and its application to control of chaos☆

Cited by 176 publications

References 11 publications

Global Hopf bifurcation analysis of an susceptible-infective-removed epidemic model incorporating media coverage with time delay

Global Hopf bifurcation analysis of an susceptible-infective-removed epidemic model incorporating media coverage with time delay

Stability and Hopf bifurcation analysis of novel hyperchaotic system with delayed feedback control

Hopf bifurcation and global stability of density‐dependent model with discrete delays involving Beddington‐DeAngelis functional response

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