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Discontinuity, Nonlinearity, and Complexity 1(2) (2012) 113--145 | DOI:10.5890/DNC.2012.04.001

Albert C. J. Luo; Jianzhe Huang

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

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Abstract

In this paper, unstable and stable period-m motions in the periodically forced Duffing oscillator are predicted analytically through the generalized harmonic balance method. Period-3, period-5 and period-7 periodic motions are investigated as examples for the Duffing oscillator with a twin-well potentials. The Hopf bifurcation of periodic motions yields the onset of period-doubling periodic motions. With increasing period number of periodic motions, there are too many co-existing stable and unstable periodic motions, and such stable periodic motions are much less than the corresponding unstable periodic motions. This investigation provides a complete picture of unstable and stable periodic motions rather than stable motions only. For any unstable periodic motion, if there is at least one co-existing stable periodic motion, then such unstable periodic motion will reach the stable periodic motion through transient motion.

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