Behaviors As Well As The Mechanism Of Different Scales In Non-autonomous Systems

The complex dynamic behaviors of nonlinear dynamic systems as well as their mechanisms have always been one of the hot topics studied by scholars at home and abroad.Based on the theory of nonlinear dynamics and the Rinzel’s fast-slow analysis method,we investigate two scales effects of two types of nonlinear non-autonomous systems,present bursting oscillations behaviors under different parameters and then analyze the mechanisms of these complex dynamic phenomena.The details are as follows.Firstly,the two scales effects and bifurcation mechanisms in the smooth system are considered.By introducing the external periodic excitation,the two scales behaviors of the three-dimensional Glukhovsky-Dolzhansky system are studied.When there is an order gap between the natural frequency and the external exciting frequency,the applied external excitation can be regarded as a slow variable,and then the original system can be regarded as a generalized autonomous system.Based on the bifurcation theory of the ordinary differential equations,the equilibrium points and bifurcation behaviors of the fast subsystem are studied.Two possible bifurcation sets,the Fold bifurcation set and the Hopf bifurcation set are obtained.Combined with the numerical simulation,the theoretical results are illustrated,and the bursting oscillation behaviors of the system are discussed under two typical parameter sets,in which we change the amplitude of the external periodic excitation while the other parameters are fixed.It is found that when the excitation amplitude increases,thevariation range of the slow-varying parameters expands and the number of pitchfork bifurcation points increases.More complicated bursting oscillations can be found in the system.Secondly,the two scales effects and bifurcation mechanisms in the non-smooth system are considered.Two scales effects of the five-dimensional non-smooth BVP system are also investigated by introducing the non-smooth factors and the external periodic excitation.With the numerical simulations,the bifurcation diagrams,phase portraits and the time history are given,which indicate that due to the existence of the non-smooth interface the behaviors of the trajectory are determined by different subsystems in different regions divided by that interface.Based on the corresponding generalized autonomous systems,the generalized equilibrium point and its stability in different regions are investigated.The multiple cross-sliding bifurcations which occur at the non-smooth interface can be derived by the theory of differential inclusion.It is shown that the movement of trajectories in different regions dominated by supercritical and subcritical Hopf bifurcations can lead to the complex bursting oscillations.