Stochastic bifurcation and nonlinear dynamics analysis of stochastic wheelset system with double time delays

Abstract

In this study, a stochastic model with dual time delays for the wheelset system is introduced. The stochastic stability and bifurcation behavior of the system, influenced by Gaussian white noise excitation, are examined, with the time delays of the primary lateral and longitudinal stiffness serving as key parameters. Initially, the central manifold theorem and stochastic averaging method are applied to reduce system dimensionality, and the system's stochastic stability is evaluated using the maximum Lyapunov exponent and singular boundary theory. Next, the conditions and forms of stochastic bifurcation are determined through the three-exponential method and joint probability density function diagrams, while the impact of the time delays of the primary lateral and longitudinal stiffness on the critical speed of stochastic P-bifurcation is analyzed. Finally, through tools such as time series plots, phase diagrams, and two-parameter bifurcation diagrams, an in-depth analysis of the system's dynamic behavior was conducted to explore how time delay affects the critical instability speed and bifurcation characteristics of the system. The simulation results indicate that an increase in the time delays of the primary lateral and longitudinal stiffness induces stochastic P-bifurcation in the system and leads to a decrease in the critical speed. The analysis of the two-parameter bifurcation diagram further reveals that, with the changes in the time delays of the primary lateral and longitudinal stiffness, the wheelset model exhibits complex periodic oscillation patterns.