# Analyzing the Stability for the Motion of an Unstretched Double Pendulum near Resonance
## 3 Analysis of the Solution  
In this section, the MST is applied to solve the EOM (5) and (6) analytically up to the third order of approximation, aiming to obtain the solvability resonance conditions and the modulation equations. To achieve this goal, the trigonometric functions \(\sin\varphi,\ \cos\varphi,\ \sin\gamma\), and \(\cos\gamma\) can be approximated up to the third order using [PERSON]'s series. Therefore, Equations (5) and (6) can be rewritten in the forms:  
\[\begin{array}{l}\ddot{\theta}_{1}+p^{2}\{\alpha\cos(pt)\ (\theta_{1}-\frac{ \theta_{1}}{6}^{3})\ -\ \beta\sin(pt)\,(1-\frac{\theta_{1}}{6}^{2})\ \}+\ \ (\theta_{1}-\frac{\theta_{1}}{6}^{3})\ +\mu\ddot{\theta}_{2}\,(1-\frac{ \theta_{2}}{2}^{2})(1-\frac{\theta_{2}}{2}^{2})\\ +\dot{\mu}\ddot{\theta}_{2}\,(\theta_{1}-\frac{\theta_{1}}{6}^{  
Thus, we can write the solutions of the second order of approximation as follows:  
\[\phi_{2}=-\frac{i\widetilde{\beta}p^{2}}{2(1-p^{2})}e^{ip\tau_{0}}+\frac{ \widetilde{\mu}\omega^{2}\text{A}_{2}}{(1-\omega^{2})}e^{i\omega\tau_{0}}+CC\, \tag{23}\]  
\[\gamma_{2}=-\frac{i\widetilde{\beta}\,\omega^{2}p^{2}}{2(\omega^{2}-p^{2})}e^{ ip\tau_{0}}+\frac{\omega^{2}\text{A}_{1}}{(\omega^{2}-1)}e^{ip\tau_{0}}+CC\,. \tag{24}\]  
Here, \(CC\) represent the preceding complex conjugates terms.  
Making use of (19), (20), (23) and (24) in Equations (17) and (18) yields secular terms. To eliminate these terms, it is required that  
\[\frac{\widetilde{\mu}\omega^{2}\text{A}_{1}}{(\omega^{2}-1)}+\frac{1}{2} \overline{\text{A}}_{1}\text{A}_{1}^{2}-2i\frac{\partial\text{A}_{1}}{\partial \tau_{2}}=0\,, \tag{25}\]  
\[\frac{\omega^{4}\widetilde{\mu}\omega^{2}\text{A}_{2}}{(1-\omega^{2})}+\frac {1}{2}\omega^{2}\overline{\text{A}}_{2}\text{A}_{2}^{2}-2i\omega\frac{ \partial\text{A}_{2}}{\partial\tau_{2}}=0\,. \tag{26}\]  
Therefore, the solutions of the third order of approximation become  
\[\begin{array}{l}\phi_{3}=\frac{\widetilde{f}_{1}}{2(1-p_{1}^{2})}e^{ip_{1} \tau_{0}}-\frac{p^{2}\widetilde{\text{A}}_{1}}{2(1-(1+p)^{2})}e^{i(1+p)\tau_{0 }}\ -\frac{p^{2}\widetilde{\text{A}}_{1}}{2(1-(1-p)^{2})}e^{i(1-p)\tau_{0}}\\ \ -\frac{\text{A}_{1}^{3}}{48}e^{3i\tau_{0}}\ -\frac{ip^{2}\widetilde{\mu} \omega^{2}}{2(1-p^{2})(\omega^{2}-p^{2})}e^{ip\tau_{0}}+CC\,\end{array} \tag{27}\]  
\[\begin{array}{l}\gamma_{3}=\frac{\widetilde{f}_{2}}{2(\omega^{2}-p_{2}^{2}) }e^{ip_{2}\tau_{0}}-\frac{p^{2}\widetilde{\text{A}}\omega^{2}\text{A}_{2}}{( 2\omega^{2}-(\omega+p)^{2})}e^{i(\omega+p)\tau_{0}}\ -\frac{\omega^{2}\text{A}_{2} ^{3}}{48}e^{3i\omega\tau_{0}}\\ \ -\frac{ip^{4}\widetilde{\mu}\omega^{2}}{2(1-p^{2})(\omega^{2}-p^{2})}e^{ip \tau_{0}}-\frac{p^{2}\widetilde{\text{A}}\text{A}_{2}}{2(\omega^{2}-(\omega-p )^{2})}e^{i(\omega-p)\tau_{0}}+CC\.\end{array} \tag{28}\]  
With the use of the following initial conditions, we can determine the unknown functions of \(A_{j}\ (j=1,2)\) from the system of the four differential equations that are represented by Equations (21), (23), (25) and (26):  
\[\begin{array}{l}\phi(0)=L_{01},\qquad\dot{\phi}(0)=L_{02},\\ \gamma(0)=L_{03},\qquad\dot{\gamma}(0)=L_{04}.\end{array}\]  
Here, \(L_{03}\ (s=1,2,3,4)\) stand for known quantities.