Multiple stability switches and Hopf bifurcation in a damped harmonic oscillator with delayed feedback
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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.
KeywordsDamped harmonic oscillator model Delayed feedback Multiple stability switches Hopf bifurcation Normal form
Mathematics Subject Classification34K09 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).
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Conflict of interest
All the authors in this manuscript declare that they have no conflict of interest.
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