This paper investigates the bifurcations of homoclinic orbits to hyperbolic saddle points in a simplified railway wheelset model with cubic and quintuple nonlinear terms. Using Melnikov’s method, the sufficient conditions for the existence of the supercritical and the subcritical pitchfork bifurcations of homoclinic orbits are proven. To determine the integrability of the variational equations around homoclinic orbits in the meaning of differential Galois theory, the corresponding Fuchsian second-order differential equation for the normal variational equation and the Riemann function are obtained. It is shown that the coefficients of the linear terms and the cubic coupling terms play a very significant role on influencing the existence of homoclinic orbits. While, the cubic coupling terms have little effect on the size of the left-hand and right-hand potential wells of homoclinic orbits. These results are beneficial to explore the key mechanism of hunting stability of a simplified railway wheelset model.

中文翻译：

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铁路轮对模型中同宿轨道分岔的存在条件

本文研究了具有三次和五元非线性项的简化铁路轮对模型中同宿轨道与双曲鞍点的分岔。利用梅尔尼科夫方法，证明了同宿轨道超临界和亚临界干草叉分岔存在的充分条件。为了确定微分伽罗瓦理论意义上的绕同宿轨道变分方程的可积性，得到了正规变分方程和黎曼函数对应的Fuchsian二阶微分方程。结果表明，线性项和三次耦合项的系数对于影响同宿轨道的存在具有非常显着的作用。而三次耦合项对同宿轨道左右势阱的大小影响不大。这些结果有利于探索简化铁路轮对模型振荡稳定性的关键机制。