# Analyzing the Stability for the Motion of an Unstretched Double Pendulum near Resonance
## 5 Steady-State Case  
The major goal of this section is to investigate the oscillations of the studied model at the steady state. The modified phases and amplitudes at the steady state can be gained from Equation (37) when the value of the derivatives of their left-hand side is zero, i.e.,  
Figure 8: Clarification of the modulation of the amplitude \(h_{2}\) as a function of \(\varphi_{2}\): (**a**) at \(\omega_{1}=(2.86,2.99,3.13)\), (**b**) at \(\omega_{2}=(4.43,4.95,5.72)\). \((\frac{d\varphi_{i}}{dt}=\frac{dh_{i}}{dt}=0;j=1,2)\)[26]. Therefore, one obtains the following algebraic system of the four equations in terms of \(h_{j}\) and \(\varphi_{j}\)\((j=1,2)\):  
\[ \begin{array}{l}h_{1}\sigma_{1}+\frac{\mu\omega^{2}h_{1}}{2(\omega^{2}-1)}+ \frac{h_{1}^{3}}{16}+\frac{f_{1}}{2}\cos\varphi_{1}=0,\\ \frac{f_{1}}{2}\sin\varphi_{1}=0,\\ h_{2}\sigma_{2}+\frac{\mu\omega^{6}h_{2}}{2(1-\omega^{2})}+\frac{\omega^{2}h_ {2}^{3}}{16}+\frac{f_{2}}{2}\cos\varphi_{2}=0,\\ \frac{f_{2}}{2\omega}\sin\varphi_{2}=0.\end{array}  \tag{38}\]  
Removing the modified phases \(\varphi_{1}\) and \(\varphi_{2}\) from (38), yields the following two algebraic equations regarding the amplitudes \(h_{j}\) and the frequency clarified by the detuning parameters \(\sigma_{j}\):  
\[\begin{array}{l}f_{1}{}^{2}=4(h_{1}\sigma_{1}+\frac{\mu\omega^{2}h_{1}}{2( \omega^  
Based on the Routh-Hurwitz criterion [27] and the following determinant:  
\[Det_{\text{k}}\left[\left( \begin{array}{cccccc}\Gamma_{1}&1&0&0&\ldots&0\\ \Gamma_{3}&\Gamma_{2}&\Gamma_{1}&1&\ldots&0\\ \Gamma_{5}&\Gamma_{4}&\Gamma_{3}&\Gamma_{2}&\ldots&0\\ \Gamma_{7}&\Gamma_{6}&\Gamma_{5}&\Gamma_{4}&\ldots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ \Gamma_{2\text{k}-1}&\Gamma_{2\text{k}-2}&\Gamma_{2\text{k}-3}&\ldots&\ldots &\Gamma_{\text{k}}\end{array} \right)\right],\]  
we can write the stability conditions at the steady-state solutions in the following forms:  
\[ \begin{array}{ll}\Gamma_{1}>0,&\Gamma_{3}(\Gamma_{1}\Gamma_{2}-\Gamma_{3})- \Gamma_{4}\Gamma_{1}{}^{2}>0,\\ \Gamma_{1}\Gamma_{2}-\Gamma_{3}>0,&\Gamma_{4}>0.\end{array}  \tag{44}\]  
Some of the resonance curves are graphed in Figures 9-11, keeping in mind the following parameter data: \(f_{1}=0.0002\), \(f_{2}=0.0003\), \(\omega=1.414\), and \(\mu=0.32\). In Figures 9 and 10, modulation amplitudes are plotted versus the detuning parameter \(\sigma_{1}\); we investigate that in the region \(-0.15\leq\sigma_{1}<0.05\) at \(\sigma_{2}=-0.005\), there exists one possible critical fixed point. This point is stable in the region \(-0.15\leq\sigma_{1}\leq-0.062\) but it loses its stability for \(-0.062<\sigma_{1}\leq 0.05\), as indicated in Figure 9. On the other hand, the stable fixed points are found in the region \(-0.15\leq\sigma_{1}\leq-0.071\) at \(\sigma_{2}=-0.05\), while it will be unstable in the area \(-0.071<\sigma_{1}\leq 0.05\), as seen in Figure 10. Moreover, Figure 11 shows the modulation amplitudes versus the detuning parameter \(\sigma_{1}\) in the region \(-0.15\leq\sigma_{1}<0.05\) at \(\sigma_{2}=0\), where only one possible critical point exists. The fixed point is stable in the region \(-0.15<\sigma_{1}\leq-0.079\) but it is unstable in the region \(-0.079<\sigma_{1}<0.05\). Figure 12: Description of the modified amplitudes via \(\tau\) when \(\omega_{1}=(2.86,2.99,3.13)\): (**a**) \(u_{1}\), (**b**) \(v_{1}\). Figure 13: Sketches of the modified amplitudes versus \(\tau\) when \(\omega_{1}=(2.86,2.99,3.13)\): (**a**) \(u_{2}\), (**b**) \(v_{2}\). possible fixed points has been checked in line with the characteristic Equation (42) and the criterion of Routh-Hurwitz (44). Time histories sketches and resonance curves have been represented graphically. The impact of the various parameters on dynamical motion has been examined using the nonlinear stability approach. The analytical results have been compared with the numerical ones of the governing system of motion to show a high degree of consistency between them, which reflects the great accuracy of the used MST.