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.
- 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

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.

References

[1]	K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A., 170 (1992), 421–428. https://doi.org/10.1016/0375-9601(92)90745-8 doi: 10.1016/0375-9601(92)90745-8
[2]	F. M. Atay, van der Pol's oscillator under delayed feedback, J. Sound Vib., 218 (1998), 333–339. https://doi.org/10.1006/jsvi.1998.1843 doi: 10.1006/jsvi.1998.1843
[3]	J. Jiang, Y. Song, Bogdanov-Takens bifurcation in an oscillator with negative damping and delayed position feedback, Appl. Math. Modell., 37 (2013), 8091–8105. https://doi.org/10.1016/j.apm.2013.03.034 doi: 10.1016/j.apm.2013.03.034
[4]	S. A. Campbell, J. Blair, T. Ohira, J. Milton, Limit cycles, tori, and complex dynamics in a second-order differential equation with delayed negative feedback, J. Dyn. Differ. Equations, 7 (1995), 213–236. https://doi.org/10.1007/BF02218819 doi: 10.1007/BF02218819
[5]	Y. Song, T. Zhang, M. O. Tad, Stability and multiple bifurcations of a damped harmonic oscillator with delayed feedback near zero eigenvalue singularity, Chaos, 18 (2008), 043113. https://doi.org/10.1063/1.3013195 doi: 10.1063/1.3013195
[6]	Z. Song, J. Xu, Codimension-two bursting analysis in the delayed neural system with external stimulations, Nonlinear Dyn., 67 (2012), 309–328. https://doi.org/10.1007/s11071-011-9979-4 doi: 10.1007/s11071-011-9979-4
[7]	J. Cao, R. Yuan, H. Jiang, J. Song, Hopf bifurcation and multiple periodic solutions in a damped harmonic oscillator with delayed feedback, J. Comput. Appl. Math., 263 (2014), 14–24. https://doi.org/10.1016/j.cam.2013.11.015 doi: 10.1016/j.cam.2013.11.015
[8]	S. A. Campbell, Y. Yuan, Zero singularities of codimension two and three in delay differential equations, Nonlinearity, 21 (2008), 2671–2691. https://dx.doi.org/10.1088/0951-7715/21/11/010 doi: 10.1088/0951-7715/21/11/010
[9]	J. Cao, R. Yuan, Multiple bifurcations in a harmonic oscillator with delayed feedback, Neurocomputing, 122 (2013), 172–180. https://doi.org/10.1016/j.neucom.2013.06.033 doi: 10.1016/j.neucom.2013.06.033
[10]	Z. Q. Qiao, X. B. Liu, D. M. Zhu, Bifurcation in delay differential systems with triple-zero singularity, Chin. Ann. Math. A., 31 (2010), 59–70.
[11]	Z. Song, J. Xu, Stability switches and Bogdanov-Takens bifurcation in an inertial two-neuron coupling system with multiple delays, Sci. China Technol. Sci., 57 (2014), 893–904. https://doi.org/10.1007/s11431-014-5536-y doi: 10.1007/s11431-014-5536-y
[12]	J. Wang, W. Jiang, Bogdanov-Takens singularity in the comprehensive national power model with delays, J. Appl. Anal. Comput., 3 (2013), 81–94. https://doi.org/10.11948/2013007 doi: 10.11948/2013007
[13]	J. Wang, X. Liu, J. Liang, Bogdanov-Takens bifurcation in an oscillator with positive damping and multiple delays, Nonlinear Dyn., 87 (2017), 255–269. https://doi.org/10.1007/s11071-016-3040-6 doi: 10.1007/s11071-016-3040-6
[14]	Y. Song, J. Jiang, Steady-state, Hopf and steady-state-Hopf bifurcations in delay differential equations with applications to a damped harmonic oscillator with delay feedback, Int. J. Bifurcation Chaos, 22 (2012), 1250286. https://doi.org/10.1142/S0218127412502860 doi: 10.1142/S0218127412502860
[15]	S. Sarwardi, S. Hossain, M. Sajid, A. S. Almohaimeed, Analysis of the Bogdanov-Takens bifurcation in a retarded oscillator with negative damping and double delay, AIMS Math., 7 (2022), 19770–19793. https://doi.org/10.3934/math.20221084 doi: 10.3934/math.20221084
[16]	J. K. Hale, Theory of Functional Differential Equations, Springer, New York, 1977. https://doi.org/10.1007/978-1-4612-9892-2
[17]	J. K. Hale, S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. https://doi.org/10.1007/978-1-4612-4342-7
[18]	Y. Xu, M. Huang, Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity, J. Differ. Equations, 244 (2008), 582–598. https://doi.org/10.1016/j.jde.2007.09.003 doi: 10.1016/j.jde.2007.09.003
[19]	T. Faria, L. T. Magalhaes, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Differ. Equations, 122 (1995), 201–224. https://doi.org/10.1006/jdeq.1995.1145 doi: 10.1006/jdeq.1995.1145
[20]	W. Jiang, Y. Yuan, Bogdanov-Takens singularity in van der Pol's oscillator with delayed feedback, Physica D, 227 (2007), 149–161. https://doi.org/10.1016/j.physd.2007.01.003 doi: 10.1016/j.physd.2007.01.003
[21]	T. Faria, L. T. Magalhaes, Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. Differ. Equations, 122 (1995), 181–200. https://doi.org/10.1006/jdeq.1995.1144 doi: 10.1006/jdeq.1995.1144
[22]	X. He, C. Li, Y. Shu, Bogdanov-Takens bifurcation in a single inertial neuron model with delay, Neurocomputing, 89 (2012), 193–201. https://doi.org/10.1016/j.neucom.2012.02.019 doi: 10.1016/j.neucom.2012.02.019
[23]	T. Dong, X. Liao, Bogdanov-Takens bifurcation in a tri-neuron BAM neural network model with multiple delays, Nonlinear Dyn., 71 (2013), 583–595. https://doi.org/10.1007/s11071-012-0683-9 doi: 10.1007/s11071-012-0683-9
[24]	M. S. Siewe, C. Tchawoua, P. Woafo, Melnikov chaos in aperiodically driven Rayleigh-Duffing oscillator, Mech. Res. Commun., 37 (2010), 363–368. https://doi.org/10.1016/j.mechrescom.2010.04.001 doi: 10.1016/j.mechrescom.2010.04.001
[25]	A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D, 16 (1985), 285–317. https://doi.org/10.1016/0167-2789(85)90011-9 doi: 10.1016/0167-2789(85)90011-9

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