Bifurcation Analysis Of Duffing System With Different Scales

Many areas in science and engineering are involved in multiple time scales.The complex bursting oscillation and its corresponding bifurcation mechanism of multiple time scales system are one of the hot topics of nonlinear science.At present,the study of smooth dynamical system has formed a complete set of mathematical theory which can describe its bifurcation behavior.However,there are lots of systems include non-smooth factors in engineering reality,such as the occurrence of impacting motion in mechanical systems,stick-slip motion in oscillations with friction and switching in electronic circuits.Therefore,it is far-reaching significant to explore non-smooth dynamical system.Based on the background above,first of all,a class of non-smooth systems with multiple time scales which take Duffing oscillator as a prototype and introduce a periodic excitation was established.The special phenomena which always behaves in the combination between large amplitude oscillation and small amplitude oscillation,namely bursting oscillation may occur when there is an order gap between the exciting frequency and the natural frequency.We choose suitable parameter values so that three different bursting oscillation were observed.Three cases known as double scrolls,three scrolls and four scrolls was considered.The mechanism of different bursting oscillation for this case of non-smooth dynamical system were presented through the transformed phase portraits analysis and bifurcation analysis,combined with non-smooth factor.By the way,aiming at the coexistence of multiple equilibrium states in such non-smooth system,it is necessary to think about the evolution of the attractor itself so that the reason for showing special oscillation phenomena in system under the influence of non-smooth factor can be explained.Next,on the basis of previous research,we set up a non-smooth system model with two excitation which combine parametric excitation and external periodic excitation.Using de Moivre formula which introduce a new slow variable v?t?and then translate parametric excitation and external periodic excitation into the function of v?t?.Thus there is only one slow variable in the system so that the mechanism of different bursting oscillation were presented by the traditional method called fast-slow analysis.In this paper,six representative groups of excited frequency were selected for comparative analysis.With help of the phase portraits and the time history to observe oscillation behavior,the corresponding laws can be revealed through combination between overlay of the bifurcation diagram and the transformed phase diagram and theoretical analysis.On the one hand,for the case when the exciting frequency is 0.01,the results show that with the increase of the ratio??₂/?₁?,the oscillation behaviors for the system become more and more complex.At the same time,the number of times for conversion when trajectory alternate the between the quiescent states and the spiking states are multiplied.On the other hand,the exciting frequency is fixed at 0.02.Finally,we concluded that in the process of introducing the slow variable,the transformed phase diagram corresponding to the system must be axisymmetric and independent of the original system as long as m and n are even number.