Bursting Oscillations And Bifurcation Mechanisms With Multiple Equilibrium States

Nonlinear dynamical systems with multiple time-scales have wide background in engineering, however, traditional nonlinear theory can’t be used to explain the mutual effect among different scales and its complex mechanism directly, as a result, a specialized theory and method is needed to be explored. So, how to deal with its complex dynamic characteristics is one of the leading subjects and hot interests of current domestic and foreign in nonlinear dynamics. The relative work is still in development stage, therefore, exploring the complex mechanism of systems with different scales not only has important scientific significance for developing nonlinear dynamical theory system, but also has vital application values for dynamic characteristics analysis, parameter identification, fault diagnosis and so on in actual engineering systems. In this thesis, the bifurcation mechanisms of bursting in the controlled systems with two time scales and the transition processes of different bursting patterns in both classic smooth and non-smooth dynamical systems with multiple equilibrium states have been investigated by using bifurcation theory of nonlinear dynamics and the fast-slow dynamic analysis. The basic contents of the thesis are given as follows.On the one hand, this paper explores the bursting behaviors as well as the mechanism of smooth systems with periodic excitation and multiple equilibrium states. Taking the controlled Lorenz model with periodic excitation as an example, the coupling effect of different scales in frequency domain corresponding to the case that an order gap exists between the exciting frequency and the natural frequency of the system with multiple equilibrium states is investigated. Unlike the autonomous slow-fast coupling systems, no obvious slow and fast subsystems can be observed in the periodically excited system. Since the exciting frequency is far less than the natural frequency of the system, the whole exciting term can be considered as a slow-varying parameter, leading to the generalized autonomous system. With the variation of the slow-varying parameter, the bifurcation forms as well as the behaviors for different equilibrium states in the generalized autonomous system are explored. It is pointed out that for certain conditions, Hopf bifurcation and fold bifurcations related to different equilibrium pointscan be observed. According to the conditions related to different bifurcations, the bursting oscillations under two typical cases are presented. In order to explore the mechanism of bursting oscillations, transformed phase portraits are introduced in which the whole exciting term is treated as a generalized state variable so that the relationship between the original state variables and the slow-varying parameter can be clearly described. By employing the transformed phase portraits, the bifurcation mechanism of different bursting attractors are presented. It is found that the coexistence of multiple equilibrium states as well as the related bifurcation forms not only leads to multiple forms of quiescent states and the spiking states, but also results in different switching forms between different quiescent states and the spiking states.On the other hand, this paper explores the bursting behaviors as well as the mechanism of piecewise-linear non-smooth systems with multiple equilibrium states and periodic excitation,especially when there are abrupt alternations between two different subsystems on both sides of non-smooth boundaries. Taking non-smooth system with piecewise-linear function and periodic excitation as an example. Phase space will be divided into several regions by non-smooth cross sections, every region is corresponding to its own linear subsystem which owns nominal equilibrium orbit, respectively. Two typical cases with different parameters in which the trajectories pass across different number of cross sections via increasing the amplitude of the excitation are investigated, resulting in different forms of bursting oscillations are obtained. It is found that the transitions between different quiescent states and spiking states may be caused by non-smooth bifurcations or the abrupt alternations between two neighboring subsystems on both sides of the non-smooth boundaries.