## 5 Conclusions  
For harmonic excitation and slight damping, stationary large amplitude solutions at subcritical excitation frequencies are obtained for the softening Duffing oscillator. These solutions have the shape of a nose in the amplitude-frequency diagram. In prior work, it has been observed that these solutions may contain large errors and that high ansatz orders may be necessary, as well as the fact that, in most cases, these solutions are unstable. The present paper aims to find descriptive explanations for this by addressing the questions given in Section 2 on the existence and accurate calculation of the nose solution as well as its stability. Those questions could be answered in the following way. With respect to the explanation of existence, the nose solutions are, in a large frequency range, a resonance phenomenon (i.e., large response compared to the excitation), in the sense that we get, for a certain excitation frequency, an amplitude which is close to the amplitude of free vibrations with same fundamental frequency. The nose solution contains a \"large portion\" of the homogeneous solution with a small addition of excitation and damping so that energy is balanced. This behavior is systematically used by methods based on Jacobian elliptic functions as reviewed in reference [20]. Therefore, we need higher ansatz orders in (classical) Harmonic Balance as the homogeneous solution contains large portions of higher harmonics (the lower the frequency, the larger the higher portions). The question on stability can simply be answered by looking at the restoring characteristics in Figure 3 left and the Harmonic Balance solutions in Figure 2. Asymptotically stable solutions can be expected in regions of positive restoring. As the amplitudes of nose solutions have almost the amplitude of the corresponding free vibrations for \(\omega~{}=~{}\Omega\), asymptotically stable solutions can (only) be expected at the right end of the nose, where, according to the larger excitation frequency and corresponding lower amplitudes, the range of positive restoring is not exceeded. Finally, as response to the question of how to alternatively calculate these solutions, a semi-analytic method considering the dominating portion of the homogeneous solution and combining Perturbation Analysis, Harmonic Balance, and residuum minimization was introduced, needing only the solution of linear algebraic equations. The method showed good results in finding the nose solutions. Figure 14: Graphical summary of the results: the gray (unstable) and black (asymptotically stable) curves show Harmonic Balance solutions with ansatz order \(n~{}=~{}1\) of the softening Duffing Equation (1) with parameter set (3). The orange line denotes the border of positive restoring. Light purple (unstable) and dark blue (asymptotically stable) dots mark the results from the Perturbation Analysis introduced in Section 4 for the larger amplitude nose solution. Green dots mark the solutions of numerical integration. **Author Contributions:** Conceptualization, U.v.W. and L.L. ; methodology, U.v.W. and L.L. ; software, L.L. (Perturbation Analysis and Harmonic Balance), N.G. (basin of attraction), and H.D. (Harmonic Balance); validation, U.v.W., L.L., H.D., and N.G. ; formal analysis, U.v.W. and L.L. ; investigation, L.L. and U.v.W. ; resources, U.v.W. and L.L. ; data curation, not applicable; writing--original draft preparation, U.v.W. ; writing--review and editing, L.L., H.D., and N.G. ; visualization, L.L., H.D., and N.G. ; supervision, U.v.W. ; project administration, U.v.W. and L.L. ; funding acquisition, L.L. and U.v.W. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)--project number LE 4694/1-1. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable.