# Research on the Vehicle Steering and Braking Stability Region
## 1 Introduction  
With the application of nonlinear dynamics in the field of vehicle dynamics, the determination of stable regions in the plane motion of vehicles has become an important research field [1]. For the vehicle system, the external input (or driver) affects the dynamic stability of the system through acceleration, deceleration, and steering, which puts the vehicle in single or coupled working conditions. Many scholars have studied the plane motion stability of vehicles under various driving conditions through different nonlinear analysis methods. Steering stability research of vehicles at a constant velocity was originally based on two-degree-of-freedom (2 DOF) nonlinear vehicle model analysis. The steering angle bifurcation caused by tire lateral force saturation has been confirmed by many scholars. The band convergence region in the phase plane has been determined as the stable region of the system, and significant results have been achieved in the stability control of lateral motion [2, 3, 4, 5]. [PERSON] et al. used the Lyapunov function method to analyze the lateral stability of vehicle steering [6], and in a follow-up study, Lyapunov exponents were also used to obtain the lateral stability region of the vehicle, which was an extension of the stability region obtained by using the Lyapunov function in the original study [7]. [PERSON] et al. combined the 2 DOF nonlinear vehicle model with a simple driver model, analyzed theHopf bifurcation characteristics of vehicles with understeer and oversteer characteristics under steady steering and found different bifurcation forms of the system through the combination of different vehicle models and driver models, and found chaos in the driving simulator as well [8; 9]. [PERSON] et al. found supercritical Hopf bifurcation of oversteer vehicles when studying the 2 DOF nonlinear model [10]. Although the research based on the classical 2 DOF nonlinear model of vehicles has achieved many research results, the model assumes that the longitudinal velocity remains unchanged and only considers the lateral force of the tires, without considering the impact of the longitudinal force of the tires on the system stability, which is not in line with the actual force situation of the tires in the actual driving of vehicles. [PERSON] et al. introduced the longitudinal velocity into the vehicle model to establish a three-degree-of-freedom (3 DOF) model and removed the assumption that the longitudinal velocity was constant [11]. Through a change of phase space trajectory, they pointed out that the essential feature of vehicle steering instability was the chaotic motion of the vehicle dynamics system. [PERSON] et al. estimated the region of attraction (ROA) of the lateral dynamics of a 3 DOF vehicle model through the sum of squares (SOS) programming algorithm [12]. [PERSON] et al. put forward an energy dissipation method of vehicle plane motion and calculated the stability region of the 3 DOF model. By comparison, it was proved that the energy dissipation method was more intuitive and easier to calculate than the Lyapunov exponents method [1]. In this study, the largest Lyapunov exponents and the sum of the Lyapunov exponents take 895,273 s (about 10 days) for numerical calculation. The total dissipation of energy takes 420 s for calculation. Based on the research on 3 DOF model steering conditions, [PERSON] et al. established a five-degree-of-freedom (5 DOF) vehicle nonlinear model incorporating driving torque, analyzed the impact of driving modes on vehicle stability under different steering angles [13], and obtained a two-dimensional bifurcation parameter set (driving stability region) under vehicle steering and driving conditions based on solving the equilibrium point of the 5 DOF system [14]. [PERSON] et al.