# Period-1 Motions and Bifurcations of a 3D Brushless DC Motor System with Voltage Disturbance
## 4 Numerical Illustrations  
A commonly used methodology for analyzing nonlinear differential equations is numerical integration, which is both straightforward to implement and efficient in computation. The analytical solution is validated through the application of numerical integration in this section. Three-dimensional displacement and frequency spectrums are utilized to depict the dynamic characteristics of the previously discussed brushless DC motor under a specific disturbance frequency. The initial condition is provided by the analytical solution. The black lines and blue circles in the figures refer to numerical solutions and analytical solutions, respectively. In the frequency spectrum, \(k=0\) presents the constant term \(a_{01}\), while \(k\
eq 0\) describes the harmonic amplitude \(A_{k/m}\). When the motion is unstable, the frequency spectrum based on a numerical solution is unreliable, so the analytical solutions are applied to plot the frequency spectrum. The value of each harmonic term is presented selectively. In the following figures, the time-history and frequency spectrum of \(x_{1}\) are presented, and 3D displacement is also illustrated. The input data for numerical illustrations are listed in Table 2.  
Figure 6a,b depicts the dynamic characteristics of stable period-1 motion when \(\Omega=6.5\). The initial conditions for numerical integration are \(x_{1}=5.3227832\), \(x_{2}=65.525293\), and \(x_{3}\)= 19.302422. The analytical solution is based on 20 harmonic terms. In 3D displacement, the analytical and numerical solutions match very well. There is one obvious loop in 3D displacement, which verifies the motion is period-1. Since the value of the constant term \(a_{01}\) is negative and the logarithmic coordinate is applied in the frequency spectrum, \(a_{01}\) is omitted while presenting the frequency spectrum. The harmonic amplitudes in frequency spectrum are \(a_{01}\approx-0.0845\), \(A_{1}\approx 6.6017\), \(A_{2}\approx 2.9657\), \(A_{3}\approx 2.5348\), \(A_{4}\approx 1.6159\), \(A_{5}\approx 0.4102\), \(A_{6}\approx 0.4759\), \(A_{7}\approx 0.1188\), \(A_{8}\approx 0.1054\), \(A_{9}\approx 0.0204\), \(A_{10}\approx 0.0131\), \(A_{11}\approx 4.5316\)\(\times 10^{-3}\), \(A_{12}\approx 3.1564\)\(\times 10^{-3}\), \(A_{13}\approx 1.2706\)\(\times 10^{-3}\), and \(A_{k/m}\)\(<1\)\(\times\)\(10^{-3}\) (\(k=14\)-\(20\); \(m=1\)). The dynamic characteristics of stable period-2 motion are depicted in Figure 5c,d when \(\Omega=6.5\). The initial conditions obtained from the analytical solution are \(x_{1}=9.7357535\), \(x_{2}=55.891501\), and \(x_{3}=13.299756\). The analytical solution for this period-2 motion is based on 40 harmonic terms. It can be inferred from Figure 6c,d that the analytical and numerical solutions match very well. The motion repeats every two excitation periods. There are two loops in 3D displacement, which verifies the period-2 motion. The harmonic amplitudes for the analytical solution are \(a_{01}\approx 0.1796\), \(A_{1/2}\approx 0.2342\), \(A_{1}\approx 6.4068\), \(A_{3/2}\approx 0.6465\), \(A_{2}\approx 3.9061\), \(A_{5/2}\approx 0.8423\), \(A_{3}\approx 1.8311\), \(A_{7/2}\approx 0.6390\), \(A_{4}\approx 1.8305\), \(A_{9/2}\approx 0.4933\), \(A_{5}\approx 0.3914\), \(A_{11/2}\approx 0.3164\), \(A_{6}\approx 0.3853\), \(A_{13/2}\approx 0.1888\), \(A_{7}\approx 0.1820\), \(A_{15/2}\approx 0.1173\), and \(A_{k}<0.05\) (\(k=16\)-\(40\), \(m=2\)). The dynamic characteristics of unstable period-1 motion are illustrated in Figure 6e,f when the disturbance frequency is 6.5. The initial conditions obtained from the analytical solution are \(x_{1}=7.5583327\), \(x_{2}=63.739675\), and \(x_{3}=18.706304\). In the first several periods, the numerical and analytical solutions match well. But the complex curves shown in 3D displacement indicate that the period-1 motion is unstable.