In this paper, we investigate the generalized Hopf bifurcation of a non-smooth railway wheelset system. It is to note that the system is a four-dimensional non-smooth differential equation. First, we show how to overcome the non-smoothness and reduce the four-dimensional system to a two-dimensional non-smooth system by the center manifold theorem. Since the two-dimensional central manifold is still non-smooth, we cannot apply the classical Hopf bifurcation theorem. Hence, we need to construct and analyze a Poincaré map so that a criterion for determining the generalized Hopf bifurcation occurring in the system is given. Finally, to demonstrate our theoretical results, we also give some numerical simulations which are presented to exhibit the corresponding bifurcation diagrams.

在本文中，我们研究了非光滑铁路轮对系统的广义Hopf分支。要注意的是，该系统是一个四维非光滑微分方程。首先，我们展示如何克服不光滑性，并通过中心流形定理将四维系统简化为二维不光滑系统。由于二维中心流形仍然不光滑，因此我们无法应用经典的Hopf分支定理。因此，我们需要构造和分析Poincaré映射，以便给出确定系统中发生的广义Hopf分支的准则。最后，为了证明我们的理论结果，我们还给出了一些数值模拟，以展示相应的分叉图。