Complex dynamics and Bogdanov-Takens bifurcations in a retarded van der Pol-Duffing oscillator with positional delayed feedback

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

doi:10.3934/mbe.2023135

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 2: 2874-2889. doi: 10.3934/mbe.2023135

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Research article Special Issues

Complex dynamics and Bogdanov-Takens bifurcations in a retarded van der Pol-Duffing oscillator with positional delayed feedback

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

Academic Editor: Jesús Martín Vaquero

Received: 08 September 2022 Revised: 06 November 2022 Accepted: 17 November 2022 Published: 01 December 2022

In this article, we will investigate a retarded van der Pol-Duffing oscillator with multiple delays. At first, we will find conditions for which Bogdanov-Takens (B-T) bifurcation occurs around the trivial equilibrium of the proposed system. The center manifold theory has been used to extract second order normal form of the B-T bifurcation. After that, we derived third order normal form. We also provide a few bifurcation diagrams, including those for the Hopf, double limit cycle, homoclinic, saddle-node, and Bogdanov-Takens bifurcation. In order to meet the theoretical requirements, extensive numerical simulations have been presented in the conclusion.

Keywords:

retarded van der Pol-Duffing oscillator,
damping,
Bogdanov-Takens bifurcation,
chaos,
center manifold,
delayed feedback,
numerical simulation

Citation: Mohammad Sajid, Sahabuddin Sarwardi, Ahmed S. Almohaimeed, Sajjad Hossain. Complex dynamics and Bogdanov-Takens bifurcations in a retarded van der Pol-Duffing oscillator with positional delayed feedback[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 2874-2889. doi: 10.3934/mbe.2023135

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Abstract

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