AI 中文总结
研究零阻尼极限下线性冲击振荡器擦边分岔，通过数值计算发现共振点间存在反复分岔序列，包括稳定周期解在不同分岔中的稳定性变化及向混沌吸引子过渡，此动力学在参数微变时持续，适用于弱阻尼冲击振荡器。
AI 中文摘要
我们考虑一个受简谐强迫的线性冲击振荡器，其冲击事件是瞬时的且有能量损失。我们研究在振荡器阻尼系数为零的极限情况下，非冲击周期解在擦边分岔处的动力学。通过数值计算表明，在共振点之间存在一系列反复出现的分岔。具体而言，共振产生一个稳定周期解，其随后在二次擦边分岔中失去稳定性，接着在鞍结分岔中恢复稳定性，然后通过倍周期级联过渡到混沌吸引子。该动力学在参数轻微变化时持续存在，适用于接近擦边的弱阻尼冲击振荡器。
英文摘要
We consider a harmonically forced linear impact oscillator, where impact events are instantaneous with energy loss. We study the dynamics at the grazing bifurcation of the non-impacting periodic solution in the limit that the damping coefficient of the oscillator is zero. Through numerical computations we show that a recurring sequence of bifurcations exists between points of resonance. Specifically, resonance creates a stable periodic solution that subsequently loses stability in a secondary grazing bifurcation, then regains stability in a saddle-node bifurcation, then transitions to a chaotic attractor through a period-doubling cascade. The dynamics persist under mild parameter variation, so apply to weakly-damped impact oscillators near grazing.
