Analysis of the Bogdanov-Takens bifurcation in a retarded oscillator with negative damping and double delay

Here we will investigate a retarded damped oscillator with double delays. We looked at the combined effect of retarded delay and feedback delay and found that the retarded delay plays a significant role in controlling the oscillation of the proposed system. Only the negative damping situation is considered in this research. At first, we will find conditions for which the origin of the proposed system becomes a Bogdanov-Takens (B-T) singularity. Also, we extract the second and the third-order normal forms of the Bogdanov-Takens bifurcation by using center manifold theory. At the end, an extensive numerical simulations have been presented to satisfy the theoretical results.