Vibrational Resonances Of Fractional-order Systems With Time Delays

Vibrational resonance refers to a phenomenon that the response amplitude of a nonlinear system at the frequency of low-frequency signal is amplified by modulating the amplitude or frequency of the high-frequency signal under the excitation of highfrequency signal and low-frequency signal.The study of vibrational resonance can not only enhance the weak signal carrying useful information,but also detect the unknown low-frequency signal.Therefore,vibrational resonance is widely used in different fields,including physics,engineering,biology,etc.The effect of time delay and the fractional order derivative on vibrational resonance is investigated in multistable system,and take the response amplitude as the index to measure the vibrational resonance phenomenon.According to the method of separation of slow and fast motions,we can obtain the system's response amplitude of the analytic expression.Combining with the Griinwald-Letnikov definition for numerical simulation of system output to verify the validity of theoretical predictions.In a fractional order quintic oscillator system with time delay feedback is studied vibrational resonance.The influence of fractional damping order and the time delay feedback on response amplitude are analysed in single-well,double-well and three-well potential,respectively.It is found that by altering the fractional order derivative,it's favour to emergence of new resonances,at the same time,fractional order can also distinguish between the resonance model.Different from the fractional order a influences on system,with the increase of time delay feedback ?,bifurcation and the equilibrium point change periodically,and this period is the period of low-frequency signal.In triple-well potential,the study found an intersection of two bifurcation points of system.The paper points out that in different steady systems,system parameters ?,?,F can be adjusted appropriately to optimum the system output.The distributed and fixed time delay were introduced into the fractional MathieuDuffing oscillator,respectively,it was found that the distributed time delay had more influence on system than the fixed time delay.In order to quantify the error between theoretical prediction and numerical simulation,this paper calculate the relative error on the response amplitude and the displacement of system.The small error shows that there is no reason not to believe that the theoretical analysis method is correct in statistical sense.In the distributed time delay model,the influence of linear stiffness coefficient and low-frequency signal of system are similar.Compared to the system without time delay,it is found that introducing time delay into model can effectively improve the system output,so as to better control the fractional Mathieu-Dufing oscillator.