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Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121--150 | DOI:10.5890/DNC.2015.06.002

Albert C.J. Luo; Yu Guo

$^{1}$ Department ofMechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL62026-1805, USA

$^{2}$ McCoy School of Engineering, Midwestern State University, Wichita Falls, TX 76308, USA

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Abstract

In this paper, periodic motions in the Duffing oscillator are investigated through the mapping structures of discrete implicit maps. The discrete implicit maps are obtained from differential equation of the Duffing oscillator. From mapping structures, bifurcation trees of periodic motions are predicted analytically through nonlinear algebraic equations of implicit maps, and the corresponding stability and bifurcation analysis of periodic motion in the bifurcation trees are carried out. The bifurcation trees of periodic motions are also presented through the harmonic amplitudes of the discrete Fourier series. Finally, from the analytical prediction, numerical simulation results of periodic motions are performed to verify the analytical prediction. The harmonic amplitude spectra are also presented, and the corresponding analytical expression of periodic motions can be obtained approximately. The method presented in this paper can be applied to other nonlinear dynamical systems for bifurcation trees of periodic motions to chaos.

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