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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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Abstract

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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