# Period-1 Motions and Bifurcations of a 3D Brushless DC Motor System with Voltage Disturbance
## 1 Introduction  
Brushless DC motors are extensively utilized in rapidly developing fields, such as electric vehicles, intelligent robots, medical equipment, aerospace, and other precision industries. However, the control system of brushless DC motors is relatively intricate and lacks excitation control. Additionally, their mechanical characteristics are rigid, particularly when the inverter output waveform is suboptimal. This can result in significant torque ripple, which adversely affects low-speed performance while increasing current loss and noise levels. Consequently, scholars have conducted extensive research on the precise control of brushless DC motors from both physical modeling and control strategy perspectives. In 1993, [PERSON] [1] proposed an approach for formulating concise representations of the nonlinear equations governing brushless DC motors, resulting in a significant reduction in the number of system control variables. [PERSON] et al. [2] established a mathematical model of brushless DC motors and proposed a model based adaptive control method. The performance of BLDC motors was analyzed and controlled through mathematical modeling by [PERSON] et al. [3]. The terminal voltage and input current were then utilized to compute the speed, torque, and flux of the sensorless BLDC motor. In addition, [PERSON] and [PERSON] [4] proposed a Matlab-based simulation model for BLDC motor drives. This model enables the effective monitoring and analysis of the dynamic characteristics of speed, torque, voltage, and current in PWM inverter components. [PERSON] et al. [5] used finite element technology to model and numerically simulate brushless permanent magnet DC motors. The aforementioned research indicates that voltage and current are crucialfactors influencing the output torque, speed control, and other performance parameters of brushless DC motors. The torque control method proposed by [PERSON] and [PERSON] [6] in 1995 was based on the back electromotive force waveform used to calculate applied output voltage, with the aim of addressing the torque ripple issue of non-ideal trapezoidal brushless DC motors. [PERSON] et al. [7] addressed the torque pulsation problem of high-performance brushless direct current motors by adjusting the phase current excitation waveform to counteract the torque pulsation. [PERSON] et al. [8] used two currents in the BLDCM phase to estimate rotor angle and velocity. [PERSON] et al. [9] proposed an adaptive Neuro-fuzzy Interference System controller based on Particle Swarm Optimization. The method was able to accurately estimate the back electromotive force at the zero-crossing point of PMBLDC motors via the terminal voltages. [PERSON] et al. [10] discussed the value of reducing torque fluctuations in brushless DC motor drive systems by studying speed, current, and commutation problems. Through the aforementioned research, it has been determined that establishing a mapping relationship between the mechanical characteristics of brushless DC motors and input parameters, such as voltage and current, can provide an enhanced technical approach for various control strategies. This holds significant importance in improving the performance and expanding the application range of brushless DC motors. However, it should be noted that brushless DC motors not only present intricate control challenges but also exhibit strong nonlinear dynamic responses under specific parameter conditions, which adversely affect their precise control. Therefore, focusing on the control model of brushless DC motors containing nonlinear terms, [PERSON] et al. [11] proposed a method based on Lyapunov stability theory to suppress and control chaotic motion according to the chaotic phenomenon of brushless DC motors under certain specific parameters. The utilization of the differential geometric control theory by [PERSON] et al. [12] facilitated the proposition of a nonlinear control approach based on current delay feedback, effectively inducing the BLDCM system to enter into a chaotic regime.