On artifact solutions of semi-analytic methods in nonlinear dynamics

Nonlinear dynamics is a topic of permanent interest in mechanics since decades. The authors have recently published some results on a very classical topic, the dynamics of a softening Duffing oscillator under harmonic excitation focusing especially on low-frequency excitation (von Wagner in Arch Appl Mech 86(8):1383–1390, 2016). In this paper, it was shown that classical tools like harmonic balance and perturbation analysis may produce artificial solutions when applied without extra carefulness with respect to parameter ranges in the case of perturbation analysis or prior knowledge about the type of solution in case of harmonic balance. In the present paper these results are shortly summarized as they give the starting point for the additional investigations described herein. First, the method of slowly changing phase and amplitude is reviewed with respect to its capability of determining asymptotic stability of stationary solutions. It is shown that this method can also produce artifact results when applied without extra carefulness. As next example an extended Duffing oscillator is investigated, which shows, if harmonic balance is applied, “islands” of solutions. Using the error criterion in harmonic balance as described in von Wagner (2016) again artifact solutions can be identified.