Static bifurcation and vibrational resonance in a fractional-order delay Duffing system with asymmetric potential are studied. By utilizing the fast and slow variable separation approach, the static bifurcation is discussed based on the equivalent system. It is found when fractional order and time delay change, the static bifurcation of fractional-order system with asymmetric potential is different from that of system with symmetric potential. In addition, changes in asymmetric parameter can also induce saddle-node bifurcation. Furthermore, according to the approximate analytical expression of response amplitude, the effects of high frequency amplitude, fractional order and asymmetric parameter on vibrational resonance are investigated. The results show that the number of resonance peaks of asymmetric potential system is related to the selection of initial value and the change of asymmetric parameter can also induce vibrational resonance. In the meantime, the good agreement between theoretical prediction and numerical simulation confirms the validity of theoretical analysis.

研究了具有非对称势的分数阶延迟Duffing系统中的静态分岔和振动共振。利用快慢变量分离方法，在等效系统的基础上讨论了静态分岔。发现当分数阶和时延变化时，非对称势分数阶系统的静态分岔与势对称系统的静态分岔不同。此外，不对称参数的变化也会引起鞍结分叉。此外，根据响应幅值的近似解析表达式，研究了高频幅值、分数阶和非对称参数对振动共振的影响。结果表明，非对称势系统共振峰的数量与初始值的选择有关，非对称参数的变化也会引起振动共振。同时，理论预测与数值模拟的良好吻合证实了理论分析的有效性。