A damped Duffing equation with a singularity is considered in this paper, where the elastic restoring force g has a singularity at origin and satisfies superlinear condition at infinity. By applying the twist theorem of nonarea-preserving map, we obtain the equation has at least one period-mT solution, where the minimal period is mT. It is also shown by bifurcation analysis that for a explicit form, the equation undergoes fold bifurcation, period doubling bifurcation and Hopf bifurcation, which leads to different solutions, including harmonic solutions, subharmonic solutions and quasiperiodic solutions. At last, we give the phase portraits and correspond Poincaré section of the solutions.

中文翻译：

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排斥型强奇异的阻尼超线性Duffing方程

本文考虑了带有奇异性的阻尼Duffing方程，其中弹性回复力g在原点具有奇异性，在无穷远处满足超线性条件。通过施加nonarea保留地图的扭转定理，我们得到方程具有至少一个周期- mT的解决方案，其中的最小周期为mT的。分叉分析还表明，对于一个显式形式，方程式经历了折叠分叉，周期加倍分叉和霍普夫分叉，从而产生了不同的解，包括谐波解，次谐波解和拟周期解。最后，给出解决方案的相图和对应的庞加莱部分。