# Nonlinear Dynamics of an Electromagnetically Actuated
## 6 Analysis of the Stability for the Primary Resonance
### Study of Stability of Steady-State Motion
The stability of the steady-state motion is determined by the eigenvalues of the Jacobian matrix and obtained from Equations (54) and (55):  
\[[J]= \begin{bmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{bmatrix}  \tag{59}\]  
where  
\[a_{11}=\frac{\partial A^{\prime}}{\partial A},\quad a_{12}= \frac{\partial A^{\prime}}{\partial B},\quad a_{21}=\frac{\partial B^{ \prime}}{\partial A},\quad a_{22}=\frac{\partial B^{\prime}}{\partial B^{ \prime}},\quad A^{\prime}=\frac{dA}{d\eta},\quad B^{\prime}=\frac{dB}{d\eta} \tag{60}\]  
After some manipulations from Equations (54) to (56) and (60), the coefficients \(a_{i\bar{\jmath}}\) take the following form:  
\[ \begin{array}{l}a_{11}=-\frac{\delta}{2}+\frac{3(k-8a_{0})}{4\Omega_{0}} \left(\frac{f}{\Omega_{0c}}\right)^{2}\frac{y^{3}}{(1+y^{2})^{2}}-15\frac{a_{ 0}}{\Omega_{0}}\left(\frac{f}{\Omega_{0c}}\right)^{4}\frac{y^{5}}{(1+y^{2})^{ 3}}-10\frac{5a_{0}}{4}\frac{a_{0}}{\Omega_{0}}\left(\frac{f}{\Omega_{0c}} \right)^{6}\frac{y^{7}}{(1+y^{2})^{4}}\\ a_{12}=-\delta+\frac{3(k-8a_{0})}{4\Omega_{0}}\left(\frac{f}{\Omega_{0c}} \right)^{2}\frac{y^{2}(1+3y^{2})}{(1+y^{2})^{2}}-15\frac{a_{0}}{\Omega_{0}} \left(\frac{f}{\Omega_{0c}}\right)^{4}\frac{y^{2}(1+5y^{2})}{(1+y^{2})^{3}}- \frac{105a_{0}}{4}\frac{a_{0}}{\Omega_{0c}}\left(\frac{f}{\Omega_{0c}}\right) ^{6}\frac{y^{7}(1+y^{2})}{(1+y^{2})^{4}}\\ a_{21}=\delta-\frac{3(k-8a_{0})}{4\Omega_{0}}\left(\frac{f}{\Omega_{0c}}\right) ^{2}\frac{y^{2}(3+y^{2})}{(1+y^{2})^{2}}+15\frac{a_{0}}{\Omega_{0}}\left( \frac{f}{\Omega_{0c}}\right)^{4}\frac{y^{2}(5+y^{2})}{(1+y^{2})^{3}}+\frac{105 }{4}\frac{a_{0}}{\Omega_{0}}\left(\frac{f}{\Omega_{0c}}\right)^{6}\frac{y^{7} }{(1+y^{2})^{4}}\\ a_{22}=-\frac{\delta}{2}-\frac{3(k-8a_{0})}{4\Omega_{0}}\left(\frac{f}{\Omega_{0c }}\right)^{2}\frac{y^{3}}{(1+y^{2})^{2}}+15\frac{a_{0}}{\Omega_{0}}\left( \frac{f}{\Omega_{0c}}\right)^{4}\frac{y^{5}}{(1+y^{2})^{3}}+\frac{105}{4}\frac {a_{0}}{\Omega_{0}}\left(\frac{f}{\Omega_{0c}}\right)^{6}\frac{y^{7}}{(1+y^{2 })^{4}}\end{array}  \tag{61}\]  
The signs of real parts of the eigenvalues of the Jacobian matrix are obtained from the following characteristic equation:  
\[\det([J]-\lambda[I_{2}])=0 \tag{62}\]  
where \([I_{2}]\) is the unity matrix of the second order and \(\lambda\) is the eigenvalue of the Jacobian matrix. Taking into account expression (6), the characteristic Equation (60) becomes the following:  
\[\lambda^{2}+tr[J]\lambda+\det[J]=0 \tag{63}\]  
where the trace of the Jacobian matrix is given by the following:  
\[tr[J]=-(a_{11}+a_{22})=c \tag{64}\]  
and the determinant \(J\) is the following:  
\[\det[J]=a_{11}a_{22}-a_{12}a_{21} \tag{65}\]  
The determinant of \(J\) is presented in Figure 13 for the above data. Figure 12: Variation in B with respect to \(\delta\). The discriminant of Equation (63) is the following:  
\[D=\left(tr[J]\right)^{2}-4\text{det}[J] \tag{66}\]  
The signs of the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) are given by  
\[\lambda_{1}=-\frac{1}{2}tr[J]\pm\frac{1}{2}\sqrt{D} \tag{67}\]  
to determine local stability. But \(c>0\), and from Figure 13, it is clear that det[\(J\)] > 0 such that the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) are real, negative, or complex with the real part negative, and therefore the steady state is asymptotically stable for the above data. Since det[\(J\)] \(\
eq 0\), there does not exist a saddle-node point, and because \(tr[J]=c\
eq 0\), Hopf bifurcation does not exist.