## 3 The Desired Solutions
\begin{array}{l}\widetilde{\chi}_{1}=B_{1}\,e^{i\tau_{0}}+\overline{B}_{1}\,e^{- i\tau_{0}},\\ \widetilde{\phi}_{1}=B_{2}\,e^{i\varpi_{1}\tau_{0}}+\overline{B}_{2}\,e^{-i\varpi_{1} \tau_{0}}-r\widetilde{\beta},\\ \widetilde{\gamma}_{1}=B_{3}\,e^{i\varpi_{2}\tau_{0}}+\overline{B}_{3}\,e^{-i \varpi_{2}\tau_{0}}+\frac{w_{1}^{2}}{w_{2}^{2}}r\widetilde{\beta};\qquad i= \sqrt{-1}\,.\end{array}  \tag{13}\]  
Here, \(B_{j}\ (j=1,2,3)\) represent functions of \(\tau_{k}\ (k=1,2)\) and \(\overline{B}_{j}\) denote their complex conjugates. Making use of the above solutions (13) into the second group of PDE (11) yields secular terms, the removal of which demands that  
\[ \begin{array}{l}D_{1}B_{1}=0,\\ D_{1}B_{2}=-\frac{i\widetilde{\beta}}{2w_{1}}r^{2}w_{1}B_{2},\\ D_{1}B_{3}=\frac{i\widetilde{\beta}}{2w_{2}}(w_{2}^{2}-w_{1}^{2})B_{3}.\end{array}  \tag{14}\]  
As a result, the second-order solutions are as follows:  
\[\begin{array}{l}\widetilde{\chi}_{2}=\frac{\varpi^{2}\lambda}{2(  
Therefore, we can convert the PDE (21) into ordinary differential equations (ODE) according to the uses of (22), (23), and the next modified phases,  
\[ \begin{array}{l}\theta_{1}(\tau_{1},\tau_{2})=\widetilde{\sigma}_{1}\tau_{1}- \psi_{1}(\tau_{1},\tau_{2}),\\ \theta_{2}(\tau_{1},\tau_{2})=\widetilde{\sigma}_{2}\tau_{1}-\psi_{2}(\tau_{1},\tau_{2}),\\ \theta_{3}(\tau_{1},\tau_{2})=\widetilde{\sigma}_{3}\tau_{1}+3\psi_{2}(\tau_{ 1},\tau_{2})-\psi_{3}(\tau_{1},\tau_{2}),\end{array}  \tag{24}\]  
into the requirement of solvability (21). Partitioning the real parts and the imaginary ones yields the next system of six first-order ODE in terms of \(a_{j}\) and \(\theta_{j}\)\((j=1,2,3)\)  
\[\begin{array}{lcl}\frac{da_{1}}{dt}&=&\frac{f_{1}}{2}\sin\theta_{1}-\frac{1} {2}c_{1}a_{1},\\ a_{1}\frac{d\theta_{1}}{dt}&=&a_  
Figure 3: Relations of \(a_{j}(\tau)\)\((j=1,2,3)\) and \(\theta_{j}(\tau)\) at \(c_{1}=0.01,\ c_{3}=0.2,\omega_{1}=3.354\) and \(\omega_{2}=3.131\): (**a–f**) when \(c_{2}(=0.018,0.021,0.024)\). Figure 4: Characterization of the time histories of \(a_{j}(\tau)\) (\(j=1,2,3\)) and \(\theta_{j}(\tau)\) at \(c_{1}=0.01,\ c_{2}=0.02\), \(\omega_{1}=3.354\), and \(\omega_{2}=3.131\): (**a–f**) when \(c_{3}(=0.19,0.2,0.21)\). The change of the various values of \(c_{2}\) is evident in the curves describing the time histories of the amplitude \(a_{2}\) and the modified phase \(\theta_{2}\) because the third and fourth equations of system (25) are dependent on \(c_{2}\), as seen in Figure 3c,d, while the other equations do not depend explicitly on \(c_{2}\). Therefore, there is no variation, to some extent, of the curves describing \(a_{1},a_{3}\) and \(\theta_{1},\theta_{3}\) as drawn in the other parts of Figure 3.  
Since the last two equations of system (25) depend on \(c_{3}\), an observed variation of the curves describes the modified phase \(\theta_{3}\), as seen in Figure 4f. There is no observed variation in the curves of the other variables because the first four equations of system (25) do not depend on \(c_{3}\) explicitly as indicated in the other parts of Figure 4.  
An examination of the system of equations (25) shows that these equations are dependent on \(\omega_{1}\) and \(\omega_{2}\). Therefore, we expect a good impact of these parameters on the time histories of \(a_{j}\)\((j=1,2,3)\) and \(\theta_{j}\) which met with the plotted curves of Figures 5 and 6. These curves describing the waves of these variables oscillate in a decaying manner as drawn in parts (a)-(d) of these figures or monotonically decrease with time as seen in parts (e) and (f) of the same figures. Based on this analysis, we come to the conclusion that the behavior of the system of equations (25) is stable and free of chaos. Figures 7-11 present the phase plane diagrams of \(a_{j}\)\((j=1,2,3)\) and \(\theta_{j}\) when \(c_{j}\) and \(\omega_{k}\)\((k=1,2)\) have various values.