Dynamical Behaviors As Well As The Bifurcation Mechanism Of Multi-scroll Systems With Two Scales In Frequency Domain

Since the development of nonlinear dynamics,scholars at home and abroad have conducted a lot of research on multi-scroll chaotic systems and achieved many fruitful results.However,there are relatively few studies on the complex behavior of multi-scroll systems with multi-scale coupling.This thesis focuses on the dynamic characteristics and evolution mechanism of multi-scroll systems with two scales in the frequency domain.In a multi-time scales coupled nonlinear dynamical system,when there is a magnitude gap between the natural frequency and the exciting frequency,the bursting oscillations formed by the combination of large amplitude oscillation and small amplitude oscillation can be observed.The multi-scroll periodic system is a system that occurs bursting oscillations many times in a cycle,which is shown as the existence of multi scroll in the corresponding phase portrait.Firstly,the dynamic characteristics of a traditional smooth multi-scroll system are investigated and a three-dimensional modified Lorenz system is taken as an example for discussion.By introducing a parametric and external excitation,a dynamic model with two scales in the frequency domain is established and the system is divided into the fast and slow subsystems.When there is a strict resonance relationship between the two excitation frequencies,the Moivre formula is used to convert the two-period excitation term into a function form of the single-period excitation term.And it is regarded as a slowly varying parameter.By changing the ratio of the two excitation frequencies and observing the overlap of equilibrium branches and transformed phase portrait,we can obtain the evolution process of the equilibrium branches and its bifurcation behavior of the fast subsystem under the slow-varying parameter.It has been found that changes in the ratio of the two excitation frequencies will induce changes in the cluster oscillation behavior of the system,corresponding to the change in the number of scrolls in the phase portrait.When the ratio of the two excitation frequencies is a non-strict resonance relationship,the product of the minimum excitation frequency and the real time is used as a slow-varying parameter to establish a generalized autonomous system for research.Research indicates that under the change of the ratio of two excitation frequencies,the system exhibits a form of alternating periodic motion and almost periodic motion as a whole.Secondly,in the research of the dynamic characteristics of non-smooth multi-scroll systems,this thesis uses a three-dimensional Filippov system as an example to discuss.External excitation is introduced as a slow-varying parameter to make the system behave as a two-scale effect in the frequency domain.The system has non-smooth terms in both the x and y directions,which makes the system form multiple interfaces.Through bifurcation analysis,the dynamic evolution characteristics of the system under the change of the amplitude of the external excitation are discussed,and the burst oscillation behavior and transition mechanism of the system at the interface due to non-smooth fold bifurcation are pointed out.At last,the research is carried out around a multi-scroll system with special nonlinear terms.This thesis uses a three-dimensional Jerk system as an example.Based on the original system,a excitation term is introduced to make the system produce the effect of the fast-slow coupling.Because the Jerk system uses a periodic function with a fixed trigonometric function as its nonlinear term,its equilibrium curve and bifurcation point have a certain periodicity.By switching the transformed phase portrait,it can be observed that there are multiple Hopf bifurcation points and fold bifurcation points on the equilibrium branches,causing the transforms between the quiescent state and spiking state,which appears as a multi-scroll phenomenon on the phase portrait.In addition,the Hopf bifurcation points in the system have no effect and they are accompanied by a delay effect under the original parameters.After adjusting the system parameters,the Hopf bifurcation points and the fold bifurcation points work together.The delay effect disappears and the number of scrolls on the corresponding phase portrait changes.