title: Stability and multiple bifurcations of a damped harmonic oscillator with delayed feedback near zero eigenvalue singularity.
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.
authors.affiliation: Department of Mathematics, Tongji University, Shanghai 200092, China.
authors.forename: Yongli
authors.initials: Y
authors.lastname: Song
authors.forename: Tonghua
authors.initials: T
authors.lastname: Zhang
authors.forename: Moses O
authors.initials: MO
authors.lastname: Tadé
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D017711:Nonlinear Dynamics; D009991:Oscillometry / Q000379:methods*