Fast-slow Dynamics Analysis Of A Coupled Duffing System With Periodic Excitation

Nonlinear dynamics have been developed rapidly in solving various problems in practical engineering and became a very active branch of mechanics in recent years.As one of the research directions of nonlinear dynamics,the multi-time scale reveals the nonlinear essential characteristics of multi-time scales from the perspective of dynamics.Its theoretical method has been widely applied to Neuroscience,Chemistry,Physics,Bioscience and other fields.In this paper,a two-degree-of-freedom nonlinear coupled Duffing equation with an external excitation and two external excitations are studied.For the coupled Duffing system with periodic excitation,the system shows the dynamic behavior on different time scales when the excitation frequency and the inherent frequency of the system are different.The first chapter introduces the research background of this paper and the research status of multi-time-scale dynamics.In the second chapter,the coupled Duffing system is studied by using the fast-slow analysis method.Firstly,we discretize the system by using the Euler method,and the discrete equation is obtained.From the time history diagram and phase diagram,we find the occurrence of bursting.Secondly,the two external excitations are considered as slow variables who are transformed into a slow variable by the Moivre formula,which divides the original system into the fast-slow subsystem.Finally,the oscillation dynamic behavior of the coupled system is discussed by combining fast-slow analysis method and the transformation phase diagram.In the third chapter,the influence of excitation frequency and excitation amplitude on the bursting is analyzed by controlling variables.Clusters will be generated when the excitation amplitude reaches a certain value.In the fourth chapter,we study the effect of time delay on the bursting with a coupled Duffing system with time delay.We find that the time delay does not seem to have effect on the occurrence of the bursting,however the upper and lower oscillation in each cycle change along with the time delay.Therefore,we can improve the peak performance by adjusting the time delay to obtain the desired peak dynamic.