Abstract
This paper undertakes an analysis of a double Hopf bifurcation of a maglev system with time-delayed feedback. At the intersection point of the Hopf bifurcation curves in velocity feedback control gain and time delay space, the maglev system has a codimension 2 double Hopf bifurcation. To gain insight into the periodic solution which arises from the double Hopf bifurcation and the unfolding, we calculate the normal form of double Hopf bifurcation using the method of multiple scales. Numerical simulations are carried out with two pairs of feedback control parameters, which show different unfoldings of the maglev system and we verify the theoretical analysis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Roger, G.: Dynamic and control requirements for EMS maglev suspension. In: Maglev 2004 Proceedings, Shanghai, China, vol. 2, pp. 926–934 (2004)
Zhang, Z., Long, Z., She, L., Chang, W.: Linear quadratic state feedback optimal control against actuator failures. In: Proceedings of the 2007 IEEE International Conference on Mechatronics and Automation, Harbin, China, pp. 3349–3354 (2007)
Zheng, J., Deng, Z.G., Wang, L.L., et al.: Stability of the maglev vehicle model using bulk high T c superconductors at low speed. IEEE Trans. Appl. Supercond. 17(2), 2103–2106 (2007)
Xiao, J.Z., Jian, J.W., Zhou, Y.H.: Effect of spring non-linearity on dynamic stability of a controlled maglev vehicle and its guideway system. J. Sound Vib. 279(1–2), 201–215 (2005)
Shi, X., She, L.: The periodic motion stability analysis of the nonlinear maglev control system. J. Dyn. Control 3, 52–55 (2005)
She, L.H., Wang, H., Zou, D.S., et al.: Hopf bifurcation of maglev system with coupled elastic guideway. In: Maglev 2008, San Diego, CA (2008)
Zou, D.S., She, L.H., Zhang, Z.Q., et al.: Maglev vehicle and guideway coupled vibration analysis. Acta Electron. Sin. 38(9), 2071–2075 (2010)
Wang, H.P., Li, J., Zhang, K.: Non-resonant response, bifurcation and oscillation suppression of a non-autonomous system with delayed position feedback control. Nonlinear Dyn. 51, 447–464 (2008)
Zhang, L.L., Huang, L.H., Zhang, Z.Z.: Hopf bifurcation of the maglev time-delay feedback system via pseudo-oscillator analysis. Math. Comput. Model. 52(5–6), 667–673 (2010)
Zhang, L.L., Campbell, S.A., Huang, L.H.: Nonlinear analysis of a maglev system with time-delayed feedback control. Physica D 240(21), 1761–1770 (2011)
Hassard, B., Kazarinoff, N., Wan, Y.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)
Wang, Z.H., Hu, H.Y.: Pseudo-oscillator analysis of scalar nonlinear time-delay systems near a Hopf bifurcation. Int. J. Bifurc. Chaos 17(8), 2805–2814 (2007)
Nayfeh, A.H.: Perturbation Methods. Shanghai Publishing House of Science and Technology, Shanghai (1984)
Nayfeh, A.H.: Method of Normal Forms. Wiley-Interscience, New York (1993)
Luongo, A., Paolone, A.: Perturbation methods for bifurcation analysis from multiple nonresonant complex eigenvalues. Nonlinear Dyn. 14, 193–210 (1997)
Luongo, A., Paolone, A., Di Egidio, A.: Multiple timescales analysis for 1:2 and 1:3 resonant Hopf bifurcations. Nonlinear Dyn. 34, 269–291 (2003)
Guckenheimer, J., Holems, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, L., Zhang, Z. & Huang, L. Double Hopf bifurcation of time-delayed feedback control for maglev system. Nonlinear Dyn 69, 961–967 (2012). https://doi.org/10.1007/s11071-011-0317-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-011-0317-7