## 4 Approximate Solution for Forced Vibrations Based on Exact Solution of Free Vibrations  
In the previous section, it was argued that the forced vibrations in the nose solutions seem to be free undamped vibrations with a small correction in amplitude and triggering of the phase, which makes it an obvious object of applying Perturbation Analysis. When determining curves like those in Figure 1 as a solution of Equation (1), Lindstedt-Poincare Perturbation Analysis is often applied, claiming that damping and excitation as well as the cubic term are small and excitation frequency \(\Omega\) and linear natural frequency \(\Omega_{0}\) are almost equal. This reflects the expectation, that the response amplitude is large compared to the excitation amplitude, which holds if damping is small and we are close to resonance. In the present case, it is still true that we have small damping and (compared to the response amplitude) small excitation amplitude. Regarding, considering the magnitude of the cubic term for the nose solutions and looking at the restoring characteristic, they have the same magnitude as the linear restoring term. So, it is not reasonable to assume this  
Figure 5: Basin of attraction for parameter set (3) and \(\Omega~{}=~{}0.5\). Following the color code from Figure 1, blue points mark initial conditions leading to the asymp-totically stable nose solution. Orange points denote initial conditions leading to the regular low-amplitude solution, while for initial conditions marked in white \(x(t)\) diverges to \(+/-\) infinity. The three points A, B, and C denote the stationary solutions at \(t=0\), cf. Appendix A.  
term to be small anymore. As we will see, application of the Lindstedt-Poincare technique, i.e., expansion of a characteristic frequency parameter, is no longer necessary here. As a result, the starting point of the analysis is the following equation:  
\[x^{\prime\prime}+\varepsilon\delta x^{\prime}+{\omega_{0}}^{2}x+\gamma x^{3}\ =\ \varepsilon \hat{f}\cos(\Omega t) \tag{17}\]  
with  
\[\delta\ =\ \frac{2D\omega_{0}}{\varepsilon};\hat{f}\ =\ \frac{f}{\varepsilon} \tag{18}\]  
and \(\varepsilon\) as a small parameter. We expand \(x(t)\) up to the first order by  
\[x\ =\ x_{0}+\varepsilon x_{1}+\ldots \tag{19}\]  
Inserting Equation (19) into Equation (17) results in  
\[x_{0}^{\prime\prime}+\varepsilon x_{1}^{\prime\prime}+\ldots+\varepsilon\delta x _{0}^{\prime}+\ldots+{\omega_{0}}^{2}(x_{0}+\varepsilon x_{1}+\ldots)+\gamma \Big{(}x_{0}^{3}+3\gamma x_{0}^{2}x_{1}+\ldots\Big{)}\ =\ \varepsilon\hat{f}\cos(\Omega t) \tag{20}\]  
Sorting with respect to order of \(\varepsilon\) gives  
\[\varepsilon^{0}:\ x_{0}^{\prime\prime}+{\omega_{0}}^{2}x_{0}+\gamma x_{0}^{3 }\ =\ 0 \tag{21}\]  
and  
\[\varepsilon^{1}:\ x_{1}^{\prime\prime}+{\omega_{0}}^{2}x_{1}+3\gamma x_{0}^{ 2}x_{1}\ =\ -\delta x_{0}^{\prime}+\hat{f}\cos(\Omega t) \tag{22}\]  
Obviously, the solution of Equation (21) is the exact solution of the free undamped vibrations discussed in the previous section. A corresponding sorting of small and non-small terms resulting in the free undamped vibrations of the Duffing oscillator or another conservative nonlinear oscillator with known exact solution as the zeroth approximation is in fact used for systems with strong nonlinearities (as we have here in the considered amplitude range). A corresponding overview is given in reference [20], describing in detail how to apply the Jacobi elliptic functions for the solution of nonlinear oscillatory problems. One example described therein is the Elliptic Harmonic Balance Method, where the Jacobi elliptic functions replacing cosine and sine as ansatz functions, and there is also a section in reference [20] dealing with systems with harmonic excitation. One paper cited therein is reference [21], which in fact considers (beside other examples), with the slightly damped and harmonically excited pendulum equation, a case very close to the case considered in the present paper.