Chaotic Motions of the van der Pol-Duffing Oscillator Subjected to Periodic External and Parametric Excitations with Delayed Feedbacks

Liang-qiang ZHOU, Fang-qi CHEN

Acta Mathematicae Applicatae Sinica(English Series) ›› 2024, Vol. 40 ›› Issue (4) : 1111-1126.

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Acta Mathematicae Applicatae Sinica(English Series) ›› 2024, Vol. 40 ›› Issue (4) : 1111-1126. DOI: 10.1007/s10255-024-1038-2
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Chaotic Motions of the van der Pol-Duffing Oscillator Subjected to Periodic External and Parametric Excitations with Delayed Feedbacks

  • Liang-qiang ZHOU, Fang-qi CHEN
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Abstract

Chaotic dynamics of the van der Pol-Duffing oscillator subjected to periodic external and parametric excitations with delayed feedbacks are investigated both analytically and numerically in this manuscript. With the Melnikov method, the critical value of chaos arising from homoclinic or heteroclinic intersections is derived analytically. The feature of the critical curves separating chaotic and non-chaotic regions on the excitation frequency and the time delay is investigated analytically in detail. The monotonicity of the critical value to the excitation frequency and time delay is obtained rigorously. It is presented that there may exist a special frequency for this system. With this frequency, chaos in the sense of Melnikov may not occur for any excitation amplitudes. There also exists a uncontrollable time delay with which chaos always occurs for this system. Numerical simulations are carried out to verify the chaos threshold obtained by the analytical method.

Key words

Van der Pol-Duffing oscillator / time delay / chaos / parametric excitation / Melnikov method

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Liang-qiang ZHOU , Fang-qi CHEN. Chaotic Motions of the van der Pol-Duffing Oscillator Subjected to Periodic External and Parametric Excitations with Delayed Feedbacks. Acta Mathematicae Applicatae Sinica(English Series), 2024, 40(4): 1111-1126 https://doi.org/10.1007/s10255-024-1038-2

References

[1] Belykh V.N., Pankratova E.V. Chaotic dynamics of two van der Pol-Duffing oscillators with Huygens coupling. Regular & Chaotic Dynamics, 15: 274-284(2010)
[2] Belykh V.N., Pankratova E.V. Shilnikov Chaos in oscillators with Huygens coupling. International Journal of Bifurcation and Chaos, 24: 1440007(2014)
[3] Cai M.X., Yang J.P., Deng J. Bifurcations and chaos in Duffing equation with damping and external excitations. Acta Mathematicae Applicatae Sinica-English Series, 30: 483-504(2014)
[4] Candido, M.R., Llibre J., Valls C. Non-existence, existence, and uniqueness of limit cycles for a generalization of the van der Pol-Duffing and the Rayleigh-Duffing oscillators. Phisica D: Nonlinear Phenomena, 407: 132458(2020)
[5] Chudzik A., Perlikowski P., Stefanski A., Kapitaniak T. Multistability and rare attractors in van der Pol-Duffing oscillator. International Journal of Bifurcation and Chaos, 21: 1907-1912(2011)
[6] Gan C.B., Lu Q.S., Huang K.L. Strongly resonant bifurcations of nonlinearly coupled van der Pol-Duffing oscillator. Applied Mathematics and Mechanics-English Edition, 20: 68-75(1999)
[7] Ghaleb A.F., Abou-Dina M.S., Moatimid G.M., Zekry M.H. Analytic approximate solutions of the cubic-quintic Duffing-van der Pol equation with two-external periodic forcing terms: stability analysis. Mathematics and Computers in Simulation, 180: 129-151(2021)
[8] Guckenheimer J., Holmes P. Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Spriger-verlag, New York (1997)
[9] Ji J.C. Nonresonant Hopf bifurcations of a controlled van der Pol-Duffing oscillator. Journal of Sound and Vibration, 297: 183-199, (2006)
[10] Ji J.C., Zhang N. Nonlinear response of a forced van der Pol-Duffing oscillator at non-resonant bifurcations of codimension two. Chaos, Solitons & Fractals, 41: 1467-1475(2009)
[11] Ji J.C., Zhang N., Wei G. Difference resonances in a controlled van der Pol-Duffing oscillator involving time delay. Chaos, Solitons & Fractals, 42: 975-980(2009)
[12] Jiang H.P., Zhang T.H., Song Y.L. Delay-induced double Hopf bifurcations in a system of two delay-coupled van der Pol-Duffing oscillators. International Journal of Bifurcation and Chaos, 25: 1550058(2015)
[13] Leung A.Y.T., Guo Z.J., Yang H.X. Fractional derivative and time delay damper characteristics in Duffingvan der Pol oscillators. Communications in Nonlinear Science and Numerical Simulation, 18: 2900-2915(2013)
[14] Liu X., Zhang T.H. Bogdanov-Takens and triple zero bifurcations of coupled van der Pol-Duffing oscillators with multiple delays. International Journal of Bifurcation and Chaos, 27: 1750133(2017)
[15] Ma S.Q., Lu Q.S., Feng Z.S. Double Hopf bifurcation for van der Pol-Duffing oscillator with parametric delay feedback control. Journalof Mathematical Analysis and Applications, 338: 993-1007, (2008)
[16] Ma X.D., Yu Y., Wang L.F. Complex mixed-mode vibration types triggered by the pitchfork bifurcation delay in a driven van der Pol-Duffing oscillator. Applied Mathematics and Computation, 411: 126522(2021)
[17] Maccari A. Vibration amplitude control for a van der Pol-Duffing oscillator with time delay. Journal of Sound and Vibration, 317: 20-29(2008)
[18] Miwadinou C.H., Monwanou A.V., Yovogan J., Hinvi L.A., Nwagonm Tuwa P.R.N., Chabi Orou J.B. Modeling nonlinear dissipative chemical dynamics by a forced modified van der Pol-Duffing oscillator with asymmetric potential: chaotic behaviors predictions. Chinese Journal of Physics, 56: 1089-1104(2018)
[19] Olabodé D.L., Miwadinou C.H., Monwanou A.V., Chabi Orou J.B. Horseshoes chaos and its passive control in dissipative nonlinear chemical dynamics. Physica Scripta, 93: 085203(2018)
[20] Pandey A., Mitra M., Ghose-Choudhury A., Guha P. On coupled delayed van der Pol-Duffing oscillators. Journal of Applied Nonlinear Dynamics, 9: 567-574(2020)
[21] Qian Y.H., Chen S.M. Accurate approximate analytical solutions for multi-degree-of-freedom coupled van der Pol-Duffing oscillators by homotopy analysis method. Communications in Nonlinear Science and Numerical Simulation, 15: 3113-3130(2010)
[22] Roy S., Das D., Banerjee D. Vibrational resonance in a bistable van der Pol-Mathieu-Duffing oscillator. International Journal of Non-linear Mechanics, 135: 103771(2021)
[23] Shen Y.J., Wen S.F., Yang S.P., Guo S.Q., Li L.R. Analytical threshold for chaos in a Duffing oscillator with delayed feedbacks. International Journal of Non-linear Mechanics, 98: 173-179(2018)
[24] Stupnicka W.S., Rudowski J. The coexistence of periodic, almost-periodic and chaotic attractors in the van der Pol-Duffing oscillator. Journal of Sound and Vibration, 199: 165-175(1997)
[25] Taffo G.I.K., Siewe M.S. Parametric resonance, stability and heteroclinic bifurcation in a nonlinear oscillator with time-delay: Application to a quarter-car model. Mechanics Research Communications, 52: 1-10(2013)
[26] Wang Y.Z., Li F.M. Dynamical properties of Duffing-van der Pol oscillator subject to both external and parametric excitations with time delayed feedback control. Journal of Vibration and Control, 21: 371-387(2015)
[27] Wen S.F., Shen Y.J., Guo S.Q. Heteroclinic bifurcation behaviors of a Duffing oscillator with delayed feedback. Shock and Vibration, 2018: 7213606(2018)
[28] Wiggers V., Rech P.C. Multistability and organization of periodicity in a van der Pol-Duffing oscillator. Chaos, Solitons & Fractals, 103: 632-637(2017)
[29] Xu J., Chung K.W. Effects of time delayed position feedback on a van der Pol-Duffing oscillator. Physica D: Nonlinear Phenomena, 180: 17-39(2003)
[30] Xu Y.Y., Luo A.C.J. Independent period-2 motions to chaos in a van der Pol-Duffing oscillator. International Journal of Bifurcation and Chaos, 30: 2030045(2020)
[31] Yu Y., Zhang Z.D., Bi Q.S. Multistability and fast-slow analysis for van der Pol-Duffing oscillator with varying exponential delay feedback factor. Applied Mathematical Modelling, 57: 448-458(2018)
[32] Yuan S.L., Jing Z.J. Bifurcations of periodic solutions and chaos in Josephson system with parametric excitation. Acta Mathematicae Applicatae Sinica-English Series, 31: 335-368(2015)
[33] Zang H., Zhang T.H., Zhang Y.D. Stability and bifurcation analysis of delay coupled van der Pol-Duffing oscillators. Nonlinear Dynamics, 75: 35-47(2014)
[34] Zhang M., Yang J.P. Bifurcations and chaos in Duffing equation. Acta Mathematicae Applicatae SinicaEnglish Series, 23: 665-684(2005)
[35] Zhu H.T. Non-stationary response of a van der Pol-Duffing oscillator under Gaussian white noise. Meccanica, 52: 833-847(2017)

Funding

The project is supported by the National Natural Science Foundation of China (No. 11772148, 12172166 and 11872201) and China Postdoctoral Science Foundation (No. 2013T60531).
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