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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


Bifurcation Trees of Period-m Motions to Chaos in a Time-Delayed, Quadratic Nonlinear Oscillator under a Periodic Excitation

Discontinuity, Nonlinearity, and Complexity 3(1) (2014) 87--107 | DOI:10.5890/DNC.2014.03.007

Albert C. J. Luo; Hanxiang Jin

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

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In this paper, analytical solutions of periodic motions in a periodi- cally excited, time-delayed, quadratic nonlinear oscillator are obtained through the Fourier series, and the stability and bifurcation of such pe- riodic motions are discussed by eigenvalue analysis. The analytical bifurcation tree of period-1 motion to chaos in such a time-delayed, quadratic oscillator is presented through period-1 to period-8 motion. Numerical illustrations of stable and unstable periodic motions are given by numerical and analytical solutions. Compared to dynami- cal systems without time-delay, the time-delayed dynamical systems possess different periodic motions and the bifurcation trees of periodic motions to chaos are also distinguishing.


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