Oscillation Behaviors Of Two Scales In A Class Of Earth Magnetic Field Model Under Periodic Excitations

The study of multi-time scale coupled problems has become one of the hot topics at home and abroad in recent years.Based on the theory of bifurcation and the method of fast and slow analysis in nonlinear dynamics in this paper,taking a kind of two time scales coupled dynamical system as an example,some related research work has been carried out,the main contents are as follows:Based on the earth magnetic reversal system model under a periodic external excitation,the effects of multiple scales in frequency domain on dynamics as well as the mechanism of the model for magnetic field reversals of the earth is deeply investigated,and the mechanism of the complex oscillation is also revealed.When the exciting frequency is far less than the natural frequency,the whole exciting term can be regarded as a slow-varying parameter,on which the original system can be considered as a generalized autonomous system,called the fast subsystem.By selecting the appropriate parameter values,then all the equilibrium branches as well as the bifurcations with the variation of the slow-varying parameter are derived.By changing the amplitude of the external periodic excitation,respectively,corresponding to different bifurcation types of generalized autonomous systems,the complex bursting oscillation modes,such as periodic Hopf/Hopf,asymmetric periodic Hopf/homo-clinic and symmetric periodic double Hopf/homo-clinic bursting oscillations are obtained.To reveal the mechanism of different types of bursting oscillation attractors,the concept of transformed phase portrait is introduced.Combining the equilibrium point bifurcation analysis with the fast subsystem,and accompanied by the gradual increase of the excitation amplitude,the mechanism of the complex dynamical evolution when the oscillation form of the system trajectories from quasi periodic oscillation to quasi periodic bursting oscillation,and then to asymmetric periodic bursting oscillation,and finally to the symmetric periodic bursting oscillation is pointed out.Similarly,still using the earth magnetic reversal system model as the originalmodel,through the simultaneous introduction of external and parametric excitation,the two time scales coupled dynamical system with an external excitation and parametric excitations are obtained.Taking the system as the object of study,we also deeply investigate the effect of multiple scales in frequency domain on dynamics as well as the mechanism of the model for magnetic field reversals of the Earth under an external excitation and parametric excitations.For two different cases,the one is an external excitation and a parametric excitation,the other is an external excitation and two parametric excitations,when there are magnitude difference between external excitation and the natural frequency of the system,the generalized autonomous system shows obvious fast and slow behavior.After determining the appropriate parameter values,combining numerical simulation,the corresponding bifurcation diagram of the equilibrium point of the generalized autonomous system is obtained.Further changing the amplitude of excitation,that is,the range of the values of the corresponding slow-varying parameter is different,then it is observed that the trajectory will pass through different equilibrium branches in a period of time,and this is an important reason leading to the occurrence of different oscillation behaviors.Through the bifurcation analysis of generalized autonomous system and the transformation phase portrait,the mechanism of bursting oscillation in different forms is further revealed.