## 2 Fundamentals of the Softening Duffing Oscillator  
The Duffing equation considered in this paper reads as follows:  
\[x^{n}(t)+2D\omega_{0}\ xt(t)+{\omega_{0}}^{2}\ x(t)+\gamma x^{3}(t)\ =\ f\cos( \varOmega t). \tag{1}\]  
Herein, \(x\) denotes the displacement, \(\omega_{0}\) the circular eigenfrequency of the linear system, \(D\) the damping ratio, \(\gamma\) the coefficient of the cubic nonlinearity, \(f\) the excitation amplitude, \(\varOmega\) the circular excitation frequency, and \(t\) the time. All these quantities are considered dimensionless, and \(()^{\prime}\) denotes the derivative with respect to dimensionless time \(t\). As we consider the softening case, \(\gamma\) is negative. In the following, the parameters \(\omega_{0}\), \(D\), \(\gamma\), and \(f\) are considered as constant in the respective parameter set, while we vary the excitation circular frequency \(\varOmega\). In the simplest case, the approximate solution of Equation (1) reads as follows:  
\[x(t)\ =\ C\cdot\cos(\varOmega t-\phi) \tag{2}\]  
with amplitude \(C\) and phase shift \(\phi\). Plotting \(C(\varOmega)\) results in the well-known behavior of the Duffing oscillator with bending of the resonance peak near \(\varOmega\ =\ \omega_{0}\) to the right in the stiffening \((\gamma>0)\) and to the left \((\gamma<0)\) in the softening case. If damping is respectively small and the excitation is large enough, a region of multiple solutions (small and large amplitude asymptotically stable, medium amplitude unstable) in the resonance peak can also be observed. This solution branch is called the regular solution in the following. A corresponding example is sketched in Figure 1 for the following parameter set:  
\[\omega_{0}\ =\ 1;\ \gamma\ =\ -0.1;\ f\ =\ 0.08;\ D\ =\ 0.022 \tag{3}\]  
i.e., a softening case with \(\gamma<0\), and the regular solution branch is sketched in orange while the results were obtained using Harmonic Balance. Figure 1: Harmonic Balance solution in Equation (2) for the Duffing Equation (1) with the parameter set (3) with the regular solution (orange), the larger amplitude nose solution (blue), and the lower amplitude nose solution (green). In addition to the regular solution with bending of the resonance peak to the left, large amplitude solutions for lower excitation circular frequencies (in the case of parameter set (3) up to \(\Omega~{}=~{}0.65\)) with the shape of a nose can be observed in the softening case only and are also found sketched in many textbooks. This solution branch will be denoted as the nose solution in the following, and the larger amplitude part is marked in blue while the lower amplitude part is marked in green in Figure 1. Validity and stability of the different solutions are discussed afterwards. In the following, we will consider some properties of these nose solutions. Therefore, we introduce higher-order solutions by Harmonic Balance with the following ansatz:  
\[x(t)~{}=~{}a_{0}+\sum_{i~{}=~{}1}^{n}a_{i}\cos(i\Omega t)+b_{i}\sin(i\Omega t) \tag{4}\]  
Ansatz (4) is introduced into Equation (1), and according to the ansatz order \(n\), the first \(n\) harmonics and the constant term are balanced while the higher-order harmonics \(n~{}+~{}1,~{}\dots,~{}3n\) (according to the cubic nonlinearity) are neglected, considering them to be small. E.g., in references [15; 16], an error criterion was introduced based on the neglected terms:  
\[\sum_{i~{}=~{}n+1}^{3n}\widetilde{a}_{i}\cos(i\Omega t)+\widetilde{b}_{i}\sin (i\Omega t) \tag{5}\]  
which can be calculated as a function of time if the coefficients \(a_{i}\), \(b_{i}\) in Equation (4) have been calculated. Based on this, in reference [15], a discussion on the errors of different types of solutions of the Duffing Equation (1) has been performed. In general, therefore the maximum in time of the absolute value of the neglected term Equation (5) is taken.