# Period-1 Motions and Bifurcations of a 3D Brushless DC Motor System with Voltage Disturbance
## 1 Introduction
The research focus of [PERSON] et al. [13] lies in the investigation of chaotic phenomena exhibited by brushless direct current motors and the exploration of strategies for achieving chaos counter control through the implementation of external controls. [PERSON] et al. [14] explored the dynamics and stability of chaos in brushless direct current motor systems. The aforementioned studies primarily address the periodic solution and stability of the brushless DC motor control model using numerical integration methods and Lyapunov stability theory. However, it is often challenging for numerical integration methods, such as the Euler method and the Runge-Kutta method, to converge at unstable solutions, thereby hindering the acquisition of complete periodic solutions. [PERSON] and [PERSON] [15; 16] used the generalized harmonic balance method to obtain analytic solutions of periodic motions in the Duffing oscillator. The asymmetric period-1 motion was obtained and bifurcation trees of period-1 motion to chaos were discussed by [PERSON] and [PERSON] [17]. [PERSON] and [PERSON] [18] presented analytical solutions for period-1 motions of a periodically forced, quadratic nonlinear oscillator. [PERSON] et al. [19] obtained symmetric period-1 motion to asymmetric period-2 motion, the independent period-2 and period-4 motions, and the corresponding bifurcation trees. [PERSON] and [PERSON] [20] investigated a one-dimensional nonlinear dynamical system. Analytical solutions of periodic motions in such 1D nonlinear dynamical systems were obtained. The frequency-amplitude characteristics of periodic motions were analyzed. [PERSON] et al. [21; 22] analyzed the same brushless DC motor model using a discrete mapping method and obtained semi-analytical periodic currents and voltages. The stability and bifurcations of the semi-analytical currents and voltages were analyzed through the eigenvalues. The feedback control of the unstable periodic motion of brushless motors with unsteady external torque was also conducted. The aforementioned methods can be taken advantage of to solve the magnetohydrodynamic [23] system and to control brushless DC motors with voltage disturbance. [PERSON] et al. [24] focused on the analytical solution of the independent periodic potentials of a 3D brushless DC motor using the generalized harmonic balance method. Their research focused more on the higher-order nonlinear potentials, but in local and solitary frequency ranges, while this study completed the research. This study obtained the complete stable and unstable nonlinear potentials of period-1. It is more important since such nonlinear potentials happen more in industrial contexts. The stable and unstable potentials and currents were obtained and the boundaries between such potentials and currents were determined. In this paper, the analytical variations of the nonlinear currents and voltages of a brushless DC motor will be studied, as well as its stability and bifurcations. The analytical solutions of the quadrature-axis voltage disturbance will be developed using the generalized harmonic balance method. The bifurcations of independent period-1 motions will be presented. Furthermore, numerical illustrations will be given to verify the analytical solutions.