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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Period-m Motions and Bifurcation Trees in a Periodically Excited, Quadratic Nonlinear Oscillator

Discontinuity, Nonlinearity, and Complexity 2(3) (2013) 263--288 | DOI:10.5890/DNC.2013.08.004

Albert C.J. Luo; Bo Yu

Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL62026-1805, USA

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In this paper, analytical bifurcation trees of period-1 motions to chaos in the periodically forced, quadratic nonlinear oscillator are discussed from the generalized harmonic balance method. The analytical solutions for stable and unstable periodic motions in such quadratic nonlinear oscillator are achieved, and the corresponding stability and bifurcation were discussed. The analytical bifurcation trees from period-1 motions to period-4 motions in such quadratic oscillator are presented, and numerical illustrations of stable and unstable periodic motions are carried out by the numerical and analytical solutions. This investigation provides a comprehensive picture of complex periodic motion in the periodically excited, quadratic nonlinear oscillator.


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