On Nonlinear Oscillations: Basins of Attraction, Harmonic Balance Solution Artifacts and Urabe's Existence Criterion for the Duffing Oscillator

ABSTRACT The Duffing oscillator is often considered as “the” prototype of a nonlinear oscillator as it exhibits many characteristic phenomena of nonlinear dynamics. One of these phenomena is the occurrence of multiple periodic solutions as considered here for the case of the harmonically excited slightly damped Duffing oscillator. As an introduction, we consider a stiffening and softening case respectively and examine the stationary solutions and their basins of attraction. In the following, the Harmonic Balance Method (HBM) is considered and applied to the softening case. The method is implemented in combination with numerical continuation to compute the frequency response of the system. In the softening case, artifacts of the HBM may arise when large amplitude solutions are considered. More than 60 years ago, Urabe provided an existence criterion for solutions of the HBM which is so far rarely applied. We demonstrate by the considered example that Urabe's criterion is capable of distinguishing artifacts from regular solutions. Some insight into the application of the criterion is given as well as a first analysis, why the criterion fails in several excitation frequency ranges and solution types.