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

Sahabuddin Sarwardi; Sajjad Hossain; Mohammad Sajid; Ahmed S. Almohaimeed; Sahabuddin Sarwardi; Sajjad Hossain; Mohammad Sajid; Ahmed S. Almohaimeed

doi:10.3934/math.20221084

AIMS Mathematics

2022, Volume 7, Issue 11: 19770-19793. doi: 10.3934/math.20221084

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Research article

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

1.
Department of Mathematics & Statistics, Aliah University, IIA/27, New Town, Kolkata-700160, India
2.
Department of Mechanical Engineering, College of Engineering, Qassim University, Buraydah-51452, Saudi Arabia
3.
Department of Mathematics, College of Science, Qassim University, Buraydah-51452, Saudi Arabia

Received: 16 May 2022 Revised: 29 August 2022 Accepted: 31 August 2022 Published: 07 September 2022
MSC : 34C15, 34K11, 34K18

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.
- retarded oscillator,
- negative damping,
- Bogdanov-Takens singularity,
- delayed feedback
Citation: Sahabuddin Sarwardi, Sajjad Hossain, Mohammad Sajid, Ahmed S. Almohaimeed. Analysis of the Bogdanov-Takens bifurcation in a retarded oscillator with negative damping and double delay[J]. AIMS Mathematics, 2022, 7(11): 19770-19793. doi: 10.3934/math.20221084

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Abstract

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.

References

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