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Stability and Hopf bifurcations of an optoelectronic time-delay feedback system

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Abstract

The local dynamics around the trivial solution of an optoelectronic time-delay feedback system is investigated in the paper, and the effect of the feedback strength on the stability is addressed. The linear stability analysis shows that as the feedback strength varies, the system undergoes exactly two times of stability switch from a stable status to an unstable status or vice versa, and at each of the two end points of the stable interval, a Hopf bifurcation occurs. To gain insight of the bifurcated periodic solution, the Lindstedt–Poincaré method that involves easy computation, rather than the center manifold reduction that involves a great deal of tedious computation as done in the literature, is used to calculate the bifurcated periodic solution, and to determine the direction of the bifurcation. Two case studies are made to demonstrate the efficiency of the method.

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References

  1. Nayfeh, A.H., Chin, C.M., Pratt, J.R.: Perturbation methods in nonlinear dynamics-applications to machining dynamics. ASME J. Manuf. Sci. Eng. 119, 485–493 (1997)

    Article  Google Scholar 

  2. Masouda, Z.N., Nayfeh, A.H.: Sway reduction on container cranes using delayed feedback controller. Nonlinear Dyn. 34, 347–358 (2004)

    Article  Google Scholar 

  3. Kyrychko, Y.N., Blyuss, K.B., Gonzalez-Buelga, A., Hogan, S.J., Wagg, D.J.: Real-time dynamic substructuring in a coupled oscillator–pendulum system. Proc. R. Soc. A 462, 1271–1294 (2006)

    MathSciNet  Google Scholar 

  4. Cai, G., Huang, J.: Optimal control method with time delay in control. J. Sound Vib. 251, 383–394 (2002)

    Article  MathSciNet  Google Scholar 

  5. Xu, X., Hu, H.Y., Wang, H.L.: Stability switches, Hopf bifurcation and chaos of a neuron model with delay-dependent parameters. Phys. Lett. A 354, 126–136 (2006)

    Article  Google Scholar 

  6. Xu, X., Hu, H.Y., Wang, H.L.: Dynamics of a two-dimensional delayed small-world network under delayed feedback control. Int. J. Bifurc. Chaos 16, 3257–3273 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ikeda, K.: Multiple-valued stationary state and its instability of the transmitted light by a ring cavity. Opt. Commun. 30, 257–261 (1979)

    Article  Google Scholar 

  8. Ikeda, K., Matsumoto, K.: High-dimensional chaotic behavior in systems with time-delay feedback. Physica D 29, 223–235 (1987)

    Article  MATH  Google Scholar 

  9. Nizette, M.: Front dynamics in a delayed feedback system with external forcing. Physica D 183, 220–224 (2004)

    Article  MathSciNet  Google Scholar 

  10. Nizette, M.: Stability of square oscillations in a delayed feedback system. Phys. Rev. E 70, 056204 (2004)

    Article  Google Scholar 

  11. Erneux, T., Larger, L., Lee, M.W., Goedgebuer, J.P.: Ikeda Hopf bifurcation revisited. Physica D 194, 49–64 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Illing, L., Gauthier, D.J.: Hopf bifurcations in time-delay systems with band-limited feedback. Physica D 210, 180–202 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Blakely, J.N., Illing, L., Gauthier, D.J.: High-speed chaos in an optical feedback system with flexible timescales. IEEE J. Quantum Electron. 40, 299–305 (2004)

    Article  Google Scholar 

  14. Faria, T., Magalhães, L.T.: Normal forms for retarded functional differential equations and applications to Bogdanov–Takens singularity. J. Differ. Equ. 122, 201–224 (1995)

    Article  MATH  Google Scholar 

  15. Faria, T., Magalhães, L.T.: Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation. J. Differ. Equ. 122, 181–200 (1995)

    Article  MATH  Google Scholar 

  16. Nayfeh, A.H.: Order reduction of retarded nonlinear systems—the method of multiple scales versus center-manifold reduction. Nonlinear Dyn. (2008). doi:10.1007/s11071-007-9237-y

    Google Scholar 

  17. Das, S.L., Chatterjee, A.: Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations. Nonliner Dyn. 30, 323–335 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Rand, R., Verdugo, A.: Hopf bifurcation formula for first order differential-delay equations. Commun. Nonlinear Sci. Numer. Simul. 12, 859–864 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Verdugo, A., Rand, R.: Hopf bifurcation in a DDE model of expression. Commun. Nonlinear Sci. Numer. Simul. 13, 235–242 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Wang, Z.H., Hu, H.Y.: Pseudo-oscillator analysis of scalar nonlinear time-delay systems near a Hopf bifurcation. Int. J. Bifurc. Chaos 17, 2805–2814 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hu, H.Y., Wang, Z.H.: Singular perturbation method for nonlinear dynamic systems with time delays. Chaos Solitons Fractals (2007). doi:10.1016/j.chaos.2007.07.048

    Google Scholar 

  22. Li, J.Y.: Hopf bifurcation of the sunflower equation. Nonlinear Anal. B: Real World Appl. (2008). doi:10.1016/j.nonrwa.2008.03.002

    Google Scholar 

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Authors and Affiliations

  1. Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, 210016, Nanjing, China

    Y. G. Zheng & Z. H. Wang

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  1. Y. G. Zheng
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  2. Z. H. Wang
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Correspondence to Z. H. Wang.

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Zheng, Y.G., Wang, Z.H. Stability and Hopf bifurcations of an optoelectronic time-delay feedback system. Nonlinear Dyn 57, 125–134 (2009). https://doi.org/10.1007/s11071-008-9426-3

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  • DOI: https://doi.org/10.1007/s11071-008-9426-3

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