Multiple stability switches and Hopf bifurcation in a damped harmonic oscillator with delayed feedback

  • Xiang-Ping YanEmail author
  • Fang-Bin Liu
  • Cun-Hua Zhang
Original paper


This paper takes into consideration a damped harmonic oscillator model with delayed feedback. After transforming the model into a system of first-order delayed differential equations with a single discrete delay, the single stability switch and multiple stability switches phenomena as well as the existence of Hopf bifurcation of the zero equilibrium of the system are explored by taking the delay as the bifurcation parameter and analyzing in detail the associated characteristic equation. Particularly, in view of the normal form method and the center manifold reduction for retarded functional differential equations, the explicit formula determining the properties of Hopf bifurcation including the direction of the bifurcation and the stability of the bifurcating periodic solutions are given. In order to check the rationality of our theoretical results, numerical simulations for some specific examples are also carried out by means of the MATLAB software package.


Damped harmonic oscillator model Delayed feedback Multiple stability switches Hopf bifurcation Normal form 

Mathematics Subject Classification

34K09 34K60 70K20 74K45 



This study was funded by the National Natural Science Foundation of China (61563026, 61763024) and Foundation of a Hundred Youth Talents Training Program of Lanzhou Jiaotong University (152022).

Compliance with ethical standards

Conflict of interest

All the authors in this manuscript declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsLanzhou Jiaotong UniversityLanzhouChina

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