Hopf bifurcation characteristics of high-speed railway bogies under regularized sliding mode control

Abstract

Sliding mode control (SMC) has been extensively investigated for the stability control of active suspension systems in high-speed trains due to its robustness and rapid response to complex disturbances. However, existing studies remain largely performance-oriented, with insufficient focus on the underlying nonlinear dynamical mechanisms and a notable lack of theoretical analysis from a bifurcation perspective. Since the Hopf bifurcation essentially determines the critical speed and the post-instability vibration characteristics, it represents a core dynamical issue for ensuring the global stability of high-speed vehicle systems. In this study, the Hopf bifurcation characteristics of a high-speed railway bogie under regularized SMC are investigated by establishing an analytical relationship between control parameters and the first Lyapunov coefficient. The analysis shows that although all SMC parameters can significantly increase the linear critical speed, their effects on the Hopf bifurcation type differ fundamentally. In particular, the proportional gain k not only enhances the linear critical speed but also converts the Hopf bifurcation from subcritical to supercritical, thereby improving the stability and safety of the system. In contrast, the switching gain k st , the sliding surface parameter c , and the regularization parameter ε , while also elevating the linear critical speed, tend to promote subcritical bifurcation behavior and thus increase the potential risk of abrupt instability. This study reveals the intrinsic connection between SMC parameters and the underlying bifurcation mechanisms, providing a bifurcation-oriented theoretical foundation for the safe design and parameter optimization of active suspension systems in high-speed railway vehicles.