We consider a single degree freedom oscillator in order to accurately represent some modeling with ship roll damping. The proposed oscillator is a nonsmooth Rayleigh–Duffing equation $\ddot{x}+a\dot{x}+b\dot{x}\left|\dot{x}\right|+cx+d{x}^3=0$. The main goal of this paper is to study the global dynamics of the nonsmooth Rayleigh–Duffing oscillator in the case $d<0$, i.e., the saddle case. The nonsmooth Rayleigh–Duffing oscillator is only $C^1$ so that many classical theory cannot be applied directly. In order to see the tendency of evolutions in a large range, we study not only its finite equilibria but also the equilibria at infinity. We find necessary and sufficient conditions for existence of limit cycles and heteroclinic loops respectively. Finally, we give the complete global bifurcation diagram and classify all global phase portraits in the Poincaré disc in global parameters.

我们考虑使用单度自由度振荡器，以便精确地表示一些带有船侧倾阻尼的模型。拟议的振荡器是一个非光滑的瑞利-达芬方程$\ddot{X}+\mathrm{一种}\dot{X}+b\dot{X}\left|\dot{X}\right|+CX+d{X}^3=0$。本文的主要目的是研究在这种情况下非光滑瑞利-达芬振荡器的整体动力学。$d<0$，即鞍形外壳。仅非光滑的瑞利-达芬振荡器$C^{\mathrm{1个}}$因此许多经典理论无法直接应用。为了观察大范围的演化趋势，我们不仅研究了其有限平衡，还研究了无穷远处的平衡。我们发现分别存在极限环和异斜环的必要和充分条件。最后，我们给出完整的全局分叉图，并按照全局参数将庞加莱圆盘中的所有全局相图分类。