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Title

Stability and multiple bifurcations of a damped harmonic oscillator with delayed feedback near zero eigenvalue singularity

Author(s)

Song, Yongli;  Zhang, Tonghua;  Tade, Moses O.

Abstract

We investigate the dynamics of a damped harmonic oscillator with delayed feedback near zero eigenvalue singularity. We perform a linearized stability analysis and multiple bifurcations of the zero solution of the system near zero eigenvalue singularity. Taking the time delay as the bifurcation parameter, the presence of steady-state bifurcation, Bogdanov-Takens bifurcation, triple zero, and Hopf-zero singularities is demonstrated. In the case when the system has a simple zero eigenvalue, center manifold reduction and normal form theory are used to investigate the stability and the types of steady-state bifurcation. The stability of the zero solution of the system near the simple zero eigenvalue singularity is completely solved.

Publication type

Journal article

Source

Chaos, Vol. 18, no. 4 (Dec 2008), article no. 043113

Publication year

2008

FOR Code(s)

0102 Applied Mathematics

Keyword(s)

Bifurcation;  Delays;  Eigenvalues and eigenfunctions;  Feedback;  Harmonic oscillators;  Nonlinear dynamical systems

Publisher

American Institute of Physics

ISSN

1054-1500

Publisher URL

http://dx.doi.org/10.1063/1.3013195

Copyright

Copyright © 2008 American Institute of Physics. Published version of this paper reproduced here in accordance with the copyright policy of the publisher.

Research Projects

Approved wavelet approaches for solving nonlinear dynamic systems in process engineering, Australian Research Council grant number DP0770420

Full text

Peer reviewed