# Analyzing the Stability for the Motion of an Unstretched Double Pendulum near Resonance
## 4 Vibrations and Conditions of Resonance  
In this section, we classify the arising different resonance cases, investigate two of them and obtain modulation equations in the framework of these cases. In general, resonance cases can be distinguished when the denominators of the above approximate solutions tend to zero [25]. Therefore, we can categorize them as a main (primary) external resonance that is found at \(p_{1}=1\ \ or\ \ p_{2}=\omega\) and an internal resonance that occurs when \(p=0\,,\ p=1\), \(p=\omega\), and \(\omega=1\). If one of the resonance cases is satisfied, such as internal resonance, we can expect that the behaviour of the dynamical system to be challenging. The above asymptotic solutions observed in the last part are right if the oscillations escape from the resonances. If any of the preceding requirements are met, that indicates the necessity to regulate the method used. Now, we are going to examine two cases of main external resonances cases that occur simultaneously. Therefore, we consider the cases \(p_{1}\approx 1\) and \(p_{2}\approx\omega\), which describe the nearness of the \(p_{1}\) and \(p_{2}\) to \(1\) and \(\omega\), respectively. Consequently, the detuning parameters \(\sigma_{j}\ (j=1,2)\) are inserted as follows:  
\[p_{1}=1+\sigma_{1},\ \ p_{2}=\omega+\sigma_{2}, \tag{29}\]then the effect of the resonance is then expressed in secular terms. The parameters of detuning are known as the distance gauge of the oscillations from the strict resonance [26]. As a result, they can be expressed in terms of \(\varepsilon\) as follows:  
\[\sigma_{j}=\varepsilon\widetilde{\sigma_{j}};\,j=1,2. \tag{30}\]  
By substituting (29) and (30) into (7) and (8), and focusing on the secular terms, the solvability conditions can be acquired from the elimination of these terms, which have the following forms:  
* For the second order approximation \[\frac{\partial\text{A}_{1}}{\partial\tau_{1}}=0,\,\,\,\,\,\,\,\,\,\frac{ \partial\text{A}_{2}}{\partial\tau_{1}}=0.\] (31)
* For the third order approximation \[\frac{\widetilde{\mu}\omega^{2}\text{A}_{1}}{\omega^{2}-1}+\frac{1}{2} \overline{\text{A}}_{1}\text{A}_{1}\text{A}_{1}^{2}-2i\frac{\partial\text{A} _{1}}{\partial\tau_{2}}+\frac{\widetilde{f}_{1}}{2}e^{i\widetilde{\sigma}_{1} \tau_{1}}=0,\] (32) \[\frac{\omega^{4}\widetilde{\mu}\omega^{2}\text{A}_{2}}{1-\omega^{2}}+\frac{1} {2}\omega^{2}\overline{\text{A}}_{2}\text{A}_{2}^{2}-2i\omega\frac{\partial \text{A}_{2}}{\partial\tau_{2}}+\frac{\widetilde{f}_{2}}{2}e^{i\widetilde{ \sigma}_{2}\tau_{1}}=0.\] (33)  
Referring to the last conditions, we can say that the observed conditions of solvability for the model constitute a nonlinear system of PDE with respect to the functions \(A_{j}\). These functions depend only upon the time scale \(\tau_{2}\). Therefore, we can write them as the following notations of polar forms:  
\[A_{j}=\frac{\widetilde{h}_{j}(\tau_{2})}{2}e^{i\widetilde{\psi}_{j}\tau_{2}}; \,h_{j}=\varepsilon\widetilde{h}_{j}\,\,\,\,\,(j=1,2). \tag{34}\]  
Here, \(\widetilde{h}_{j}\) and \(\psi_{j}(j=1,2)\) represent the real functions of amplitudes and phases of \(\varphi\) and \(\gamma\), respectively, while \(h_{i}\) (\(i=1,2\)) are the amplitudes of the generalized coordinates \(\phi\) and \(\gamma\) according to the hypotheses (9) and (10). It is clear that \(A_{j}\) is an independent function of \(\tau_{0}\) and \(\tau_{1}\). Therefore, we can write the first order derivative as follows:  
\[\frac{dA_{j}}{d\tau}=\varepsilon^{2}\frac{\partial A_{j}}{\partial\tau_{2}}; \,(j=1,2). \tag{35}\]  
Based on the above formula, we can turn the PDE Equations (32) and (33) into ordinary differential equations (ODE). Then, the modified phases can be represented according to  
\[\varphi_{j}(\tau_{1},\tau_{2})=\tau_{j}\widetilde{\varphi_{j}}-\psi_{j}(\tau_ {2});\,(j=1,2). \tag{36}\]  
By making use of (34)-(36) in (32) and (33), and then separating the real parts and imaginary ones, we can obtain the following:  
\[