# Period-1 Motions and Bifurcations of a 3D Brushless DC Motor System with Voltage Disturbance
## 2 Analytical Solutions
\tag{13}\]  
From Equation (13), Equation (12) becomes  
\[\dot{\mathbf{x}}=\mathbf{F}(\mathbf{x},t). \tag{14}\]  
From [11; 12; 13], the appropriate solution for period-\(m\) motion from Equation (14) is  
\[ \begin{split} x_{1}^{*}&=a_{01}^{(m)}(t)+\sum\limits_{k=1}^ {N}b_{k/m1}(t)\cos\frac{k}{m}\Omega t+c_{k/m1}(t)\sin\frac{k}{m}\Omega t\\ x_{2}^{*}&=a_{02}^{(m)}(t)+\sum\limits_{k=1}^{N}b_{k/m2}(t)\cos \frac{k}{m}\Omega t+c_{k/m2}(t)\sin\frac{k}{m}\Omega t\\ x_{3}^{*}&=a_{03}^{(m)}(t)+\sum\limits_{k=1}^{N}b_{k/m3}(t)\cos \frac{k}{m}\Omega t+c_{k/m3}(t)\sin\frac{k}{m}\Omega t\end{split}  \tag{15}\]  
where \(a_{0i}^{(m)}(t),b_{k/mi}(t),c_{k/mi}(t)\) are the Fourier-alike coefficients. Taking the derivative of Equation (15) with respect to time gives  
\[ \begin{split}\dot{x}_{1}^{*}&=\dot{a}_{01}^{(m)}+ \sum\limits_{k=1}^{N}\,\left(\dot{b}_{k/m1}+\frac{k}{m}\Omega c_{k/m1}\right) \cos\frac{k}{m}\Omega t+&\left(\dot{c}_{k/m1}-\frac{k}{m}\Omega b _{k/m1}\right)\sin\frac{k}{m}\Omega t\\ x_{2}^{*}&=\dot{a}_{02}^{(m)}+\sum\limits_{k=1}^{N} \,\left(\dot{b}_{k/m2}+\frac{k}{m}\Omega c_{k/m2}\right)\cos\frac{k}{m}\Omega t +&\left(\dot{c}_{k/m2}-\frac{k}{m}\Omega b_{k/m2}\right)\sin\frac{ k}{m}\Omega t\\ \dot{x}_{3}^{*}&=\dot{a}_{03}^{(m)}+\sum\limits_{k=1}^{N} \,\left(\dot{b}_{k/m3}+\frac{k}{m}\Omega c_{k/m3}\right)\cos\frac{k}{m}\Omega t +&\left(\dot{c}_{k/m3}-\frac{k}{m}\Omega b_{k/m3}\right)\sin\frac{ k}{m}\Omega t\end{split} \quad. \tag{16}\]  
With the substitution of Equations (15) and (16) into Equation (14), and averaging all terms of constants, \(\cos\left(k\Omega t/m\right)\) and \(\sin\left(k\Omega t/m\right)\) yield  
\[ \begin{split}\dot{a}_{01}^{(m)}&=F_{01}^{(m)},\,\dot {a}_{02}^{(m)}=F_{02}^{(m)},\,\dot{a}_{03}^{(m)}=F_{03}^{(m)};\\ \dot{b}_{k/m1}&=-\frac{k}{m}\Omega c_{k/m1}+F_{01}^{( m)},\,\dot{b}_{k/m2}=-\frac{k}{m}\Omega c_{k/m2}+F_{02}^{(m)},\,\dot{b}_{k/m3}=- \frac{k}{m}\Omega c_{k/m3}+F_{03}^{(m)};\\ \dot{c}_{k/m1}&=\frac{k}{m}\Omega b_{k/m1}+F_{21}^{( m)},\,\dot{c}_{k/m2}=\frac{k}{m}\Omega b_{k/m2}+F_{22}^{(m)},\,\dot{c}_{k/m3}= \frac{k}{m}\Omega b_{k/m3}+F_{23}^{(m)}.\end{split}  \tag{17}\]  
Let  
\[ \begin{split}\mathbf{a}_{0}^{(m)}&=(a_{01}^{(m)},a_{0 2}^{(m)},a_{03}^{(m)})^{\mathrm{T}},\\ \mathbf{b}^{(m)}&=(\mathbf{b}_{1/m^{\prime}},b_{2/m^{ \prime}},\cdots,b_{N/m})^{\mathrm{T}},\mathbf{b}_{k/m}=(b_{k/m1^{\prime}},b_{k/ m2},b_{k/m3})^{\mathrm{T}}\\ \mathbf{c}^{(m)}&=(c_{1/m^{\prime}}c_{2/m^{\prime}}, \cdots,c_{N/m})^{\mathrm{T}},\mathbf{c}_{k/m}=(c_{k/m1^{\prime}},c_{k/m2},c_{k/ m3})^{\mathrm{T}}\\ \dot{\mathbf{a}}_{0}^{(m)}&=(\dot{a}_{01}^{(m)},\dot{a }_{02}^{(m)},a_{03}^{(m)})^{\mathrm{T}}\\ \dot{\mathbf{b}}^{(m)}&=(\dot{\mathbf{b}}_{1/m},\dot{ \mathbf{b}}_{2/m},\cdots,\dot{\mathbf{b}}_{N/m})^{\mathrm{T}},\dot{\mathbf{b}} _{k/m}=(\dot{b}_{k/m1},\dot{b}_{k/m2},\dot{b}_{k/m3})^{\mathrm{T}}\\ \dot{\mathbf{c}}^{(m)}&=(\dot{\mathbf{c}}_{1/m},\dot{ \mathbf{c}}_{2/m},\cdots,\dot{\mathbf{c}}_{N/m})^{\mathrm{T}},\dot{\mathbf{c}}_{k /m}=(\dot{c}_{k/m1},\dot{c}_{k/m2},\dot{c}_{k/m3})^{\mathrm{T}}\\ \mathbf{k}_{1}&=diag[\mathbf{I}_{3\times 3},2\mathbf{I}_{3 \times 3},\cdots,N\mathbf{I}_{3\times 3}]\end{split}  \tag{18}\]  
and for \(k=1,2,\ldots,N\)  
\[ \begin{split}\mathbf{F}_{0}^{(m)}(\mathbf{a}_{0}^{(m)},\mathbf{b} ^{(m)},\mathbf{c}^{(m)})&=\frac{1}{mT}\int_{0}^{mT}\mathbf{F}( \mathbf{x}^{*},t)dt,\\ \mathbf{F}_{1k}^{(m)}(\mathbf{a}_{0}^{(m)},\mathbf{b}^{(m)},\mathbf{c }^{(m)})&=\frac{2}{mT}\int_{0}^{mT}\mathbf{F}(\mathbf{x}^{*},t) \cos(\frac{k}{m}\Omega t)dt,\\ \mathbf{F}_{2k}^{(m)}(\mathbf{a}_{0}^{(m)},\mathbf{b}^{(m)},\mathbf{c }^{(m)})&=\frac{2}{mT}\int_{0}^{mT}\mathbf{F}(\mathbf{x}^{*},t) \sin(\frac{k}{m}\Omega t)dt,\end{split}  \tag{19}\]where  
\[