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<p>This paper establishes a dynamic model of the bogie system on curved tracks and investigates the system's hunting stability, multistable dynamics, and codimension-2 bifurcations. The stability of the system is assessed by using eigenvalues of the linear Jacobian matrix, yielding the evolution of the linear critical speed with key parameter changes. By applying center manifold theory and projection method, the first Lyapunov coefficient is computed to classify the type of Hopf bifurcation (supercritical or subcritical). Although many previous studies have effectively predicted the hunting instability of bogie systems through Hopf bifurcation theory, the multistability between saddle-node bifurcation and Hopf bifurcation after instability has not been sufficiently studied. Therefore, we reveal three types of hysteresis behavior associated with Hopf bifurcations of the bogie system, and analyze the evolution of multistability, critical speed, and uncertainty with respect to key parameter variations. The results reveal that when the degree of uncertainty is zero, a degenerate Hopf bifurcation occurs. A cusp bifurcation of limit cycle occurs when two saddle-node bifurcation curves coincide. A Hopf-Hopf bifurcation occurs when two Hopf bifurcation curves intersect. When the first Lyapunov coefficient is equal to zero, the system undergoes a degenerate Hopf bifurcation, also known as the generalized Hopf bifurcation (Bautin bifurcation). The topological structures near the generalized Hopf bifurcation points are thoroughly analyzed under different parameter conditions, and the mechanisms of multistability in their vicinity are revealed. Furthermore, the local dynamical behavior of the Hopf-Hopf bifurcation point is investigated, revealing the existence of a quasi-periodic three-dimensional torus between the two Neimark-Sacker bifurcation curves. Finally, the Hopf bifurcation types in the two-parameter plane are predicted and their evolution laws are analyzed. The results obtained in this paper can provide certain theoretical reference for the parameter optimization of vehicles.</p>

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Codimension-2 bifurcations and multistability of a bogie system on curved tracks

  • 期刊论文

  • 发表日期：

    2025年9月4日

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  • Wen Zhang,
  • Pengcheng Miao,
  • Yuan Yue

摘要

This paper establishes a dynamic model of the bogie system on curved tracks and investigates the system's hunting stability, multistable dynamics, and codimension-2 bifurcations. The stability of the system is assessed by using eigenvalues of the linear Jacobian matrix, yielding the evolution of the linear critical speed with key parameter changes. By applying center manifold theory and projection method, the first Lyapunov coefficient is computed to classify the type of Hopf bifurcation (supercritical or subcritical). Although many previous studies have effectively predicted the hunting instability of bogie systems through Hopf bifurcation theory, the multistability between saddle-node bifurcation and Hopf bifurcation after instability has not been sufficiently studied. Therefore, we reveal three types of hysteresis behavior associated with Hopf bifurcations of the bogie system, and analyze the evolution of multistability, critical speed, and uncertainty with respect to key parameter variations. The results reveal that when the degree of uncertainty is zero, a degenerate Hopf bifurcation occurs. A cusp bifurcation of limit cycle occurs when two saddle-node bifurcation curves coincide. A Hopf-Hopf bifurcation occurs when two Hopf bifurcation curves intersect. When the first Lyapunov coefficient is equal to zero, the system undergoes a degenerate Hopf bifurcation, also known as the generalized Hopf bifurcation (Bautin bifurcation). The topological structures near the generalized Hopf bifurcation points are thoroughly analyzed under different parameter conditions, and the mechanisms of multistability in their vicinity are revealed. Furthermore, the local dynamical behavior of the Hopf-Hopf bifurcation point is investigated, revealing the existence of a quasi-periodic three-dimensional torus between the two Neimark-Sacker bifurcation curves. Finally, the Hopf bifurcation types in the two-parameter plane are predicted and their evolution laws are analyzed. The results obtained in this paper can provide certain theoretical reference for the parameter optimization of vehicles.