# Period-1 Motions and Bifurcations of a 3D Brushless DC Motor System with Voltage Disturbance
## 2 Analytical Solutions
\begin{split}\mathbf{F}_{0}^{(m)}&=(F_{01}^{(m)},F_{02}^{ (m)},F_{03}^{(m)})^{\text{T}}\\ \mathbf{F}_{1}^{(m)}&=(\mathbf{F}_{11}^{(m)},F_{12}^{ (m)},\cdots,\mathbf{F}_{1N}^{(m)})^{\text{T}}\\ \mathbf{F}_{2}^{(m)}&=(\mathbf{F}_{21}^{(m)},F_{22}^{ (m)},\cdots,\mathbf{F}_{2N}^{(m)})^{\text{T}}\\ \mathbf{F}_{1k}^{(m)}&=(F_{1k1}^{(m)},F_{1k2}^{ (m)},F_{1k3}^{(m)})^{\text{T}}\\ \mathbf{F}_{2k}^{(m)}&=(F_{2k1}^{(m)},F_{2k2}^{ (m)},F_{2k3}^{(m)})^{\text{T}}\end{split}  \tag{20}\]  
and \(F_{01}^{(m)}\), \(F_{02}^{(m)}\), \(F_{03}^{(m)}\), \(F_{1k1}^{(m)}\), \(F_{1k2}^{(m)}\), \(F_{1k3}^{(m)}\), \(F_{2k1}^{(m)}\), \(F_{2k2}^{(m)}\), and \(F_{2k3}^{(m)}\) are presented in Appendix A. From Equations (18)-(20), Equation (17) becomes  
\[ \begin{split}\dot{\mathbf{a}}_{0}^{(m)}&=\mathbf{F }_{0}^{(m)}(\mathbf{a}_{0}^{(m)},\mathbf{b}^{(m)},\mathbf{c}^{(m)}),\\ \dot{\mathbf{b}}^{(m)}&=-\Omega\frac{\mathbf{k}_{1} }{m}\mathbf{c}^{(m)}+\mathbf{F}_{1}^{(m)}(\mathbf{a}_{0}^{(m)},\mathbf{b}^{( m)},\mathbf{c}^{(m)}),\\ \dot{\mathbf{c}}^{(m)}&=\Omega\frac{\mathbf{k}_{1} }{m}\mathbf{b}^{(m)}+\mathbf{F}_{2}^{(m)}(\mathbf{a}_{0}^{(m)},\mathbf{b}^{( m)},\mathbf{c}^{(m)}).\end{split}  \tag{21}\]  
Introduce  
\[\mathbf{z}^{(m)}=(\mathbf{a}_{0}^{(m)},\mathbf{b}^{(m)},\mathbf{c}^{(m)})^{ \text{T}},\mathbf{f}^{(m)}=(\mathbf{F}_{0}^{(m)},-\Omega\frac{\mathbf{k}_{1} }{m}\mathbf{c}^{(m)}+\mathbf{F}_{1}^{(m)},\Omega\frac{\mathbf{k}_{1}}{m} \mathbf{b}^{(m)}+\mathbf{F}_{2}^{(m)})^{\text{T}}. \tag{22}\]  
The standard form of Equation (22) is  
\[\dot{\mathbf{z}}^{(m)}=\mathbf{f}^{(m)}(\mathbf{z}^{(m)}). \tag{23}\]  
The periodic motion of Equation (12) can be obtained by Equation (23) with \(\dot{\mathbf{z}}^{(m)}=0\). The equilibrium \(\mathbf{z}^{(m)*}\) is obtained by solving the nonlinear algebraic equation of \(\mathbf{f}^{(m)}(\mathbf{z}^{(m)*})=\mathbf{0}\). In the neighborhood of \(\mathbf{z}^{(m)*}\), \(\mathbf{z}^{(m)}=\mathbf{z}^{(m)*}+\Delta\mathbf{z}^{(m)}\), the linearized equation of Equation (23) is  
\[\Delta\dot{\mathbf{z}}^{(m)}=D\mathbf{f}^{(m)}(\mathbf{z}^{(m)*})\Delta\mathbf{ z}^{(m)} \tag{24}\]  
and the Jacobian matrix is  
\[D\mathbf{f}^{(m)}(\mathbf{z}^{(m)*})=\left.\frac{\partial\mathbf{f}^{(m)}}{ \partial\mathbf{z}^{(m)}}\right|_{\mathbf{z}^{(m)*}} \tag{25}\]  
And  
\[ \begin{split}\frac{\partial\mathbf{f}^{(m)}}{\partial\mathbf{z} ^{(m)}}&=\left[\begin{matrix}\frac{\partial\mathbf{F}_{0}^{(m)}}{ \partial\mathbf{a}_{0}^{(m)}}&\frac{\partial\mathbf{F}_{0}^{(m)}}{ \partial\mathbf{b}^{(m)}}&\frac{\partial\mathbf{F}_{0}^{(m)}}{\partial \mathbf{c}^{(m)}}\\ \frac{\partial\mathbf{F}_{1}^{(m)}}{\partial\mathbf{a}_{0}^{(m)}}& \frac{\partial\mathbf{F}_{1}^{(m)}}{\partial\mathbf{b}^{(m)}}&-\Omega \frac{\mathbf{k}_{1}}{m}+\frac{\partial\mathbf{r}_{1}^{(m)}}{\partial \mathbf{c}^{(m)}}\\ \frac{\partial\mathbf{F}_{2}^{(m)}}{\partial\mathbf{a}_{0}^{(m)}}& \Omega\frac{\mathbf{k}_{1}}{m}+\frac{\partial\mathbf{r}_{1}^{(m)}}{\partial \mathbf{b}^{(m)}}&\frac{\partial\mathbf{r}_{1}^{(m)}}{\partial \mathbf{c}^{(m)}}\end{matrix}\right]=\left[\begin{matrix}\mathbf{G}_{0}(3 \times(6N+3))\\ \mathbf{G}_{1}(3N\times(6N+3))\\ \mathbf{G}_{2}(3N\times(6N+3))\end{matrix}\right]\end{split}  \tag{26}\]  
where  
The bifurcations of period-1 motions, as the disturbance frequency varies within the range of \(\Omega\) (1.218, 1.247), are depicted in Figure 3. In generalized harmonic balance theory, when the disturbance frequency is close to zero, the analytical solution requires infinite harmonic terms to approximate the motion. The presence of a low disturbance frequency necessitates a larger number of harmonic terms. Therefore, analytic solutions with 80 harmonic terms are applied when analyzing the bifurcation plots in this disturbance frequency range. There are three independent period-1 motions in this range. For the first independent motion in the range of \(\Omega\) (1.218,1.244), the critical frequencies of saddle-node bifurcations are \(\Omega\) = 1.2196 and 1.2222, and no Hopf bifurcation is observed.