Nonlinear Dynamics

, Volume 88, Issue 4, pp 2831–2862 | Cite as

Bifurcation trees of period-3 motions to chaos in a time-delayed Duffing oscillator

Original Paper
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Abstract

In this paper, bifurcation trees of period-3 motions to chaos in a periodically forced, time-delayed, hardening Duffing oscillator are investigated by a semi-analytical method. This comprehensive investigation provides a way to looking into the period-m motions to chaos in nonlinear dynamical systems. Such a semi-analytical method is based on the differential equation discretization of the time-delayed, nonlinear dynamical system. Bifurcation trees for the stable and unstable solutions of period-3 motions to chaos in such a time-delayed, Duffing oscillator are obtained semi-analytically. From the finite discrete Fourier series, harmonic frequency–amplitude curves for stable and unstable solutions of period-3 to period-6 motions are developed for quantity levels, singularity and catastrophes of harmonic amplitudes. From the analytical prediction, numerical simulations of period-3 and period-6 motions in the time-delayed Duffing oscillator are carried out which are compared with the corresponding analytical predictions. Through numerical illustrations, the complexity and asymmetry of period-3 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. Based on the quantity levels of harmonic amplitudes, effects of the corresponding harmonics on the periodic motions can be identified.

Keywords

Time-delayed Duffing oscillator Period-3 motions to chaos Bifurcation tree Implicit mapping Mapping structures Nonlinear frequency–amplitudes 

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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial EngineeringSouthern Illinois University EdwardsvilleEdwardsvilleUSA

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