## 2 Fundamentals of the Softening Duffing Oscillator
This error criterion was also applied, e.g., in reference [16] and other publications of the first two authors and it may be normalized, e.g., by the maximum of the corresponding solution \(x(t)\). Other error criteria are studied, e.g., in reference [17]. The general outcome of the analysis of the solutions for low ansatz orders, e.g., in reference [15] is that, beside the solutions sketched in Figure 1 with zero mean value, there exist solutions with nonzero mean value, i.e., \(a_{0}\
eq 0\). These unstable solutions represent forced vibrations around the nonzero equilibrium positions of the softening Duffing oscillator. These solutions are not considered in the present paper. Considering the solutions with zero mean value, it is found that the regular solution contains, in general, small errors, while the nose solutions show large errors. This resulted in the assumption that these solutions can be considered as artifacts, e.g., in reference [15], as the fundamental assumption of the Harmonic Balance method--neglected terms are small--is not fulfilled. Therefore, it can be expected that the solution calculated by low-order Harmonic Balance differs largely from the sought-after solution of the differential equation and is therefore only a solution of the method but not of the considered system. In fact, later investigations of the problem, e.g., in reference [18] show that higher orders in the Harmonic Balance are necessary to make the nose solutions converge and that the shape of the nose, its amplitudes, and its frequency range of existence vary in parts largely compared to the low-order solution in Figure 1. Based on the methods in reference [18], Figure 2 shows corresponding results (in order to improve readability, the number of plotted points has been drastically reduced by the Ramer-Douglas-Peucker algorithm with a max. distance threshold of \(10^{-5}\)). In reference [18], several a posteriori measures are discussed in order to detect and characterize the artifact behavior. For this, a mathematical (Supplementary Materials) definition of artifact solutions is introduced and several residual, geometric, algebraic, and solver-related error measures are studied. The utilized solvers are the so-called algebraic HBM in conjunction with a quasi-Gauss-Newton method for the numerical path following the frequency response curves. While, in many textbooks on nonlinear vibrations, the low-order nose solutions (e.g., \(n~{}=~{}1\)), as in Figure 1, are sketched without further considerations, as well as the lower part of the nose being denoted as unstable and the upper part as asymptotically stable, a general stability analysis using Floquet theory shows that most parts of the nose are unstable. This can also be observed in Figure 2 in part A, where only small parts of the upper part of the nose solution close to the nose tip are asymptotically stable, while all other parts are unstable. Whether there exist asymptotically stable nose parts depends on the respective parameter set. While the considered parameter set in reference [18] does not show any asymptotically stable parts, there are asymptotically stable parts in the parameter set considered in reference [19] and as mentioned in the parameter set (3), considered here as a first example. Based on the analysis so far, we want to address the following questions in this paper:  
1. Is there a descriptive, respectively ostensive explanation for the existence of the nose solution and is there also a descriptive explanation for why these solutions are partly hard to calculate in Harmonic Balance? 2. How can we explain the fact that only (small) parts of the nose solutions are asymptotically stable, depending on the parameter set, while sometimes there are no stable parts in these isolated solutions? 3. Can the corresponding insights be used for a semi-analytic method of calculating the nose solutions? The authors are aware that the addressed problem (i.e., the Duffing equation) is more than 100 years old and has been subject of numerous examinations.