Bifurcation trees of period-1 motions in a periodically excited, softening Duffing oscillator with time-delay

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In this paper, bifurcation trees of period-1 motions to chaos in a periodically excited, time-delay, softening Duffing oscillator are analytically predicted through an implicit mapping method. Discretization of the time-delay oscillator gives an implicit mapping. Stable and unstable periodic motions in such a time-delay, softening Duffing oscillator are achieved through the corresponding mapping structures. From the finite discrete Fourier series, harmonic frequency–amplitude characteristics for stable and unstable solutions of period-1 to period-4 motions are developed, and the singularity, catastrophes and quantity levels of harmonic amplitudes are presented. A symmetric period-1 motion with symmetric break generates three branches of period-1 motions to chaos. From the analytical prediction, periodic motions in the time-delay softening Duffing oscillator are simulated numerically. The bifurcation trees of period-1 motions to chaos in the time-delay softening Duffing oscillator are difficult to be obtained from the traditional analytical methods.