# Resonance in the Cart-Pendulum System--An Asymptotic Approach
## 4 System's Stability  
The aim of this research is to look into the system's stability using Equations (11) and (12) and through investigating the simultaneous primary external resonance \(p=1\), \(p=K\) and one of the internal resonance cases \(K=1,3K=1\) at the second approximation. Therefore, the detuning parameters \(\sigma_{j}\ \ (j=1,2)\) are considered in the following way [32]:  
\[p\approx 1+\sigma_{1},\ \ \ 1\approx K+\sigma_{2}. \tag{28}\]  
Substituting (28) into (16) and (17) and omitting the secular terms, the following solvability conditions are obtained for the second-order approximations:  
\[-2 iD_{2}A+\frac{\hat{c}_{1}^{2}A}{4}+\frac{\hat{\mu}^{2}\alpha A}{(K^{2}-1)}(1 +2B\overline{B}K^{2})+\frac{1}{2}\hat{f}e^{i\sigma_{1}T_{1}}=0, \tag{29}\]  
\[2 iKD_{2}B-\frac{\hat{c}_{2}^{2}B}{4}+\frac{\hat{\mu}^{2}K^{4}\alpha B}{(K^{2}- 1)}(1+B\overline{B})+\frac{i\alpha\hat{\mu}}{(K^{2}-1)}(\hat{c}_{2}-\hat{c}_{1 }K^{2})Ae^{i\sigma_{2}T_{1}}=0. \tag{30}\]  
We can analyze the above two equations through expressing \(A\) and \(B\) in the following polar forms:  
\[A=\frac{1}{2}a(T_{2})e^{i\delta(T_{2})},\ \ \ B=\frac{1}{2}b(T_{2})e^{i\gamma(T_{2})}, \tag{31}\]  
where \(a,b\) and \(\delta,\gamma\) represent the amplitudes of the motions and their corresponding phases. \[ \begin{array}{c}\theta_{1}(T_{1},T_{2})=\hat{\sigma}_{1}(T_{1})-\delta(T_{2 }),\\ \theta_{2}(T_{1},T_{2})=\hat{\sigma}_{2}(T_{1})+\delta(T_{2})-\gamma(T_{2}); \\ \sigma_{j}=\hat{\epsilon}\hat{\sigma}_{j}\ \ \ (j=1,2).\end{array}  \tag{32}\]  
Let us define the following modified phases. Substitution of (31) and (32) into (29) and (30) yields  
\[ \begin{array}{c}-i\hat{a}+a\hat{\delta}+\frac{1}{8}a\hat{c}_{1}^{2}+\frac{a \alpha\hat{\mu}^{2}}{4(K^{2}-1)}(2+b^{2}K^{2})+\frac{\hat{f}}{2}(\cos\theta_{1 }+i\sin\theta_{1})=0,\\ -iK\hat{b}+b\gamma+\frac{1}{8}b\hat{c}_{2}^{2}-\frac{a\hat{\mu}}{8(K^{2}-1)}[ \hat{\mu}K^{4}b(b^{2}+4)+4 ia(\hat{c}_{2}-K^{2}\hat{c}_{1})\\ \qquad\qquad\qquad\qquad\qquad\times(\cos\theta_{2}+i\sin\theta_{2})]=0.\end{array}  \tag{33}\]  
Multiplying (33) by \(\hat{\epsilon}^{2}\), then using (10) and (32) and separating the real and imaginary parts in each side of the resultant equations, we obtain the following modulation equations of the amplitudes and phases:\[a\dot{\theta}_{1}=a\sigma_{1}+\frac{ac_{1}^{2}}{8}+\frac{\mu^{2}aa}{4(K^{2}-1)}[2 +(Kb)^{2}]+\frac{1}{2}f\cos\theta_{1},\] \[\dot{a}=\frac{1}{2}f\sin\theta_{1},\] \[\dot{b}=\frac{a\mu a}{2K(K^{2}-1)}(K^{2}c_{1}-c_{2})\cos\theta_{2}, \tag{34}\] \[b\dot{\theta}_{2}=b(\sigma_{1}+\sigma_{2})-b\dot{\theta}_{1}+ \frac{bc_{2}^{2}}{8}-\frac{a\mu}{2(K^{2}-1)}[\mu bK^{4}(\frac{b^{2}}{4}+1)\] \[+a(K^{2}c_{1}-c_{2})\sin\theta_{2}],\]  
where dots are the differentiation with respect to \(\tau\). Focusing on the previous system of Equation (34), we can see that it identifies the amplitudes \(a,b\) and the modified phases \(\theta_{1},\theta_{2}\) of the investigation of the considered simultaneous resonances cases. For the steady motion, we have \(\dot{a}=\dot{b}=\dot{\theta}_{1}=\dot{\theta}_{2}=0\)[33], which correspond to the equilibrium points of system (34). Therefore, one obtains  
\[f=\frac{c_{1}^{2}}{4}+2a\sigma_{1}+\frac{\mu^{2}\alpha a}{2(K^{2}-1)}[2+(Kb)^ {2}, \tag{35}\]  
\[\frac{a\mu\alpha}{2(K^{2}-1)}(c_{2}-K^{2}c_{1})=\frac{\mu^{2}K^{4}ab}{8(K^{2} -1)}(b^{2}+4)-\frac{c_{2}^{2}b}{8}-b(\sigma_{1}+\sigma_{2}). \tag{36}\]  
These equations are solved numerically using the Newtonian method [34] through Matlab programs [21; 35] to represent the relation between the amplitudes \(a\) and \(b\) graphically in order to obtain the possible steady-state solutions close to the resonance. Since the investigated resonances appear simultaneously, then Equations (35) and (36) should be considered as a set of nonlinear equations relative to the variables \(a\) and \(b\). It is worth mentioning that the numbers of possible amplitudes (solutions) range from at least one to seven at most. This variation is strongly dependent on the considered parameters, as can be seen in Figures 2-4.