Bogdanov–Takens bifurcation in an oscillator with positive damping and multiple delays

In this paper, a damped harmonic oscillator with multiple delays is investigated. The existence conditions under which the origin of the system is a Bogdanov–Takens (B–T) singularity are derived. By utilizing the center manifold reduction and choosing suitable bifurcation parameters, the second- and the third-order normal forms of the B–T bifurcation for the system are obtained. Finally, numerical simulations are presented to illustrate the theoretical criteria.

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