## 3 Exact Solution for Free Undamped Vibrations and Conclusions for Forced Vibrations
Therefore, the nose solution seems (at least for parameter sets comparable to (3)) to contain a \"large portion\" of free vibration, i.e., the homogeneous solution (however, with a well-defined phase shift determined by the phase of the external excitation) with a small addition of excitation and damping, so that energy is balanced. In fact, in many textbooks, \"backbone\" curves with the relation \(\omega(A)\) (mostly based on the method used for the determination of the forced vibration, e.g., Harmonic Balance) are also sketched in the curves of forced vibrations like \[\frac{1}{2}x^{\prime 2}+\frac{1}{2}\omega_{0}x^{2}+\frac{1}{4}\gamma x^{4}\ =\ E_{0}\ =\ \text{const}. \tag{7}\]0 As a restriction, it should be mentioned that this seems to be valid for large parts of the nose but not for excitation circular frequencies close to zero. As can be seen in Figure 2 for \(\Omega\) smaller than 0.1 in the higher-order Harmonic Balance solutions, the amplitude increases for decreasing \(\Omega\) and significantly exceeds the maximum possible amplitude value (for parameter set (3)) of the free undamped solutions \(\sqrt{10}~{}\approx~{}3.1623\). Also, the second part of question 1--why is it so hard to calculate this solution by Harmonic Balance--can therefore be answered. At least the \"large portion\" of free vibration contains large portions of higher harmonics (the lower the frequency the larger the portions) which ultimately makes it necessary to use high ansatz orders in Harmonic Balance. Looking at the restoring characteristics in Figure 3 left and the Harmonic Balance solutions in Figure 2, there is also a very simple descriptive explanation for why only (small) parts of the nose solutions are asymptotically stable, depending on the parameter set, while there are often no stable parts in these isolated solutions, i.e., the response to question 2. Asymptotically stable solutions can be expected in regions of positive restoring, i.e., any solution exceeding the amplitude of \(\sqrt{10}\) for parameter set (3) is suspicious for becoming unstable. As the amplitudes of nose solutions have almost the amplitude of the corresponding free vibrations for \(\omega~{}=~{}\Omega\), plus an amplitude correction depending on excitation amplitude and damping, asymptotically stable solutions can be expected only at the right end of the nose, where, according to larger excitation frequency and following the previous argumentation corresponding lower amplitudes, the range of positive restoring is not exceeded. This behavior can be observed exactly when testing several parameter sets. Parameter set (3) is in fact chosen with small damping and excitation frequency so that the nose exists (i.e., it is not merged with the bent resonance peak) but turns out to reach enough to the right, that sufficiently small amplitudes result that there is a positive restoring characteristic. Figure 5 shows the basin of attraction of the asymptotically stable solutions. To determine the basin of attraction, Equation (1) with the parameter set (3) and \(\Omega~{}=~{}0.5\) is numerically integrated for various initial conditions until a stationary solution is reached. The initial conditions used are uniformly distributed over the range shown below. In order to accelerate the large number of numerical integrations, these were executed in parallel on the GPU (Nvidia RTX 4090). Since this is significantly faster with single precision, a custom-optimized integration kernel was written. This uses the Runge Kutta4 th algorithm with a fixed step size. In convergence studies and comparison calculations with the integrator ODE45 implemented in Matlab R2024a, the time step size is adapted to \(h~{}=~{}2~{}\pi/100\Omega~{}=~{}T/100\) so that the results are in sufficient alignment. In Figure 5, orange points denote initial conditions leading to the regular low-amplitude solution (as in Figure 1), while for initial conditions marked in white, \(x(t)\) diverges to \(+/-\) infinity.