Nonlinear Dynamics

, Volume 93, Issue 3, pp 1277–1286 | Cite as

One-to-three resonant Hopf bifurcations of a maglev system

  • Lingling Zhang
  • J. C. Ji
Original Paper


This paper studies the dynamics of a maglev system around 1:3 resonant Hopf–Hopf bifurcations. When two pairs of purely imaginary roots exist for the corresponding characteristic equation, the maglev system has an interaction of Hopf–Hopf bifurcations at the intersection of two bifurcation curves in the feedback control parameter and time delay space. The method of multiple time scales is employed to drive the bifurcation equations for the maglev system by expressing complex amplitudes in a combined polar-Cartesian representation. The dynamics behavior in the vicinity of 1:3 resonant Hopf–Hopf bifurcations is studied in terms of the controller’s parameters (time delay and two feedback control gains). Finally, numerical simulations are presented to support the analytical results and demonstrate some interesting phenomena for the maglev system.


Maglev system Multiple scales Resonant Hopf–Hopf bifurcations 



This work is supported by Natural Science Foundation of Hunan Province (2018JJ2192), the Scientific Research Key Project of Hunan Provincial Education Department (16A106) and the China Scholarship Council (CSC) in 2017.

Compliance with ethical standards

Conflict of interest

The authors declare that for this article, there is no conflict of interest in authorial ascription to organizations or financial and personal relationships with other people.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Information TechnologyHunan Women’s UniversityChangshaPeople’s Republic of China
  2. 2.School of Mechanical and Mechatronic EngineeringUniversity of Technology SydneyBroadwayAustralia

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