Abstract
In this chapter, time-delay effects on periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are reviewed and further discussed. Bifurcation trees of periodic motions to chaos varying with time-delay are presented for such a time-delayed, Duffing oscillator. From the analytical prediction, periodic motions in the time-delayed, hardening Duffing oscillator are simulated numerically. Through numerical illustrations, time-delay effects on period-1 motions to chaos in nonlinear dynamical systems are strongly related to the distributions and quantity levels of harmonic amplitudes.
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References
Hu, H.Y., Dowell, E.H., Virgin, L.N.: Resonance of harmonically forced Duffing oscillator with time-delay state feedback. Nonlinear Dyn. 15(4), 311–327 (1998)
Hu, H.Y., Wang, Z.H.: Dynamics of Controlled Mechanical Systems with Delayed Feedback. Springer, Berlin (2002)
MacDonald, N.: Harmonic balance in delay-differential equations. J. Sound Vib. 186(4), 649–656 (1995)
Leung, A.Y.T., Guo, Z.: Bifurcation of the periodic motions in nonlinear delayed oscillators. J. Vib. Control 20, 501–517 (2014)
Liu, L., Kalmar-Nagy, T.: High-dimensional harmonic balance analysis for second-order delay-differential equations. J. Vib. Control 16(7–8), 1189–1208 (2010)
Luo, A.C.J.: Analytical solutions of periodic motions in dynamical systems with/without time-delay. Int. J. Dyn. Control 1, 330–359 (2013)
Luo, A.C.J., Jin, H.X.: Bifurcation trees of period-m motion to chaos in a time-delayed, quadratic nonlinear oscillator under a periodic excitation. Discontinuity Nonlinearity Complexity 3, 87–107 (2014)
Luo, A.C.J., Jin, H.X.: Complex period-1 motions of a periodically forced Duffing oscillator with a time-delay feedback. Int. J. Dyn. Control 3(4), 325–340 (2015)
Luo, A.C.J., Jin, H.X.: Period-\(m\) motions to chaos in a periodically forced Duffing oscillator with a time-delay feedback. Int. J. Bifurcat. Chaos 24(10), article no: 1450126, 20 p (2014)
Luo, A.C.J.: Periodic flows in nonlinear dynamical systems based on discrete implicit maps. Int. J. Bifurcat. Chaos 25(3), Article No: 1550044, 62 p (2015)
Luo, A.C.J., Guo, Yu.: A semi-analytical prediction of periodic motions in Duffing oscillator through mapping structures. Discontinuity Nonlinearity Complexity 4(2), 121–150 (2015)
Luo, A.C.J., Xing, S.Y.: Symmetric and asymmetric period-1 motions in a periodically forced, time-delayed, hardening Duffing oscillator. Nonlinear Dyn. 85, 1141–1186 (2016)
Luo, A.C.J., Xing, S.Y.: Multiple bifurcation trees of period-1 motions to chaos in a periodically forced, time-delayed, hardening Duffing oscillator. Nonlinear Dyn. 89, 405–434 (2016)
Luo, A.C.J., Xing, S.Y.: Time-delay effects on periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator. J. Vib. Test. Syst. Dyn. 1(1), 73–91 (2017)
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Luo, A.C.J., Xing, S. (2018). Time-Delay Effects on Periodic Motions in a Duffing Oscillator. In: Edelman, M., Macau, E., Sanjuan, M. (eds) Chaotic, Fractional, and Complex Dynamics: New Insights and Perspectives. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-68109-2_5
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DOI: https://doi.org/10.1007/978-3-319-68109-2_5
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