Nonlinear Dynamics

, Volume 49, Issue 1–2, pp 117–129 | Cite as

Stability, bifurcation and chaos of a delayed oscillator with negative damping and delayed feedback control

  • X. Xu
  • H. Y. Hu
  • H. L. Wang
Original Article


This paper presents a detailed analysis on the dynamics of a delayed oscillator with negative damping and delayed feedback control. Firstly, a linear stability analysis for the trivial equilibrium is given. Then, the direction of Hopf bifurcation and stability of periodic solutions bifurcating from trivial equilibrium are determined by using the normal form theory and center manifold theorem. It shows that with properly chosen delay and gain in the delayed feedback path, this controlled delayed system may have stable equilibrium, or periodic solutions, or quasi-periodic solutions, or coexisting stable solutions. In addition, the controlled system may exhibit period-doubling bifurcation which eventually leads to chaos. Finally, some new interesting phenomena, such as the coexistence of periodic orbits and chaotic attractors, have been observed. The results indicate that delayed feedback control can make systems with state delay produce more complicated dynamics.


Time delay Delayed spring force Hopf bifurcation Chaos Period-doubling bifurcation 


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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.College of MathematicsJilin UniversityChangchunP.R. China
  2. 2.Institute of Vibration Engineering ResearchNanjing University of Aeronautics and AstronauticsNanjingP.R. China

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