Study On Mechanism Of A Nonlinear Systems With Different Scales Effect

Multi-scale effect of non smooth systems is one of the important research topic in recent years,which gets more attention of scholars at home and abroad.This article mainly explores the bursting oscillation mechanism of the second non smooth systems under the multi-scale effect.In the process of the bursting oscillation,the dynamic behavior and the unconventional bifurcation of system trajectory are researched in non smooth interface.The bursting oscillation phase diagram under different set of parameters is given through the numerical simulations,presenting the oscillation phenomenon with different modes in non smooth interface,using the bifurcation theory to explain the mechanism of transition between different modes,and analyzes the influence of unconventional bifurcation for the complex dynamic behavior of the system mechanism.Firstly,taking chaotic geomagnetic field model as an example,a new Filippov system is constructed by introducing non-smooth factors and increasing periodic excitation terms.By analyzing the stability of the equilibrium points in each region,the possible conditions for fold bifurcations and Hopf bifurcations in the process of bursting oscillations are given.With the help of the differential inclusion theory,unconventional bifurcations are analyzed for the trajectory in the non-smooth interface.It is found that the non smooth bifurcation will affect the structure of the bursting attractor in the spiking state,and cause a special transition from spiking state to silent state.In addition,the change of system parameters will lead to the chaotic phenomenon of the oscillation.Secondly,by taking the Filippov type which is formed according to the Lü system,we do the analysis of equilibrium branches as well as bifurcations.All the parameters is fixed,except the bifurcation parameter b.By means of changing the bifurcation parameter b,dynamic behaviors of trajectory on the non-smooth interface,can be obtained with numerical simulation.All types of bifurcations on the interface have been investigated via numerical computation with bifurcation theory.Hence,the bursting oscillation as well as its mechanism in one period is revealed by overlapping the equilibrium branches with the transformed phase portrait.