自激作用下洛伦兹振子的簇发现象及其分岔机制

在表现为稳定极限环的自激振子作用下的洛伦兹系统，在不同时间尺度下具有特殊的非线性现象．通过快子系统的平衡点及其特性分析，给出了快子系统随激励强度变化的分岔条件，分析了系统随激励强度变化的动力学演化过程，指出当激励强度增长到一定程度并满足快子系统产生fold分岔条件时，系统会产生fold／fold簇发，其中沉寂态表现为快子系统的平衡态，激发态为围绕快子系统焦点的振荡．讨论了其相应的簇发机制，并进一步揭示了簇发现象随参数发生变化的过程，随着激励强度的继续增加，虽然簇发定性保持不变，但在两对称的激发态的接近旋转中心处，系统会沿着快子系统的平衡态来回运动，其长度近似等于fold分岔点与激励项幅值之间的距离．

Bursting phenomenon as well as the bifurcation mechanism of self-excited Lorenz system

The Lorenz system under the influence of self-excited oscillator, which behaves in stable periodic limit cycle, may exhibit rich special nonlinear phenomena when different time scales are introduced. Bifurcation conditions of the fast subsystem have been derived via the analysis of the equilibrium points as well as the corresponding characteristics. Upon the investigation of the dynamical evolution of coupled the system, it is pointed out that, fold/fold bursting may appear when the excitation increases to meet with the conditions of fold bifurcation of the fast subsystem, in which the quiescent state corresponds to the equilibrium state of the fast subsystem, while the spiking state is related to the fluctuation around the loci of the fast subsystem. The bifurcation mechanism is presented, which is applied to the exploration of the bursting evolution with the variation of the parameters. It is found that, with the increase of the excitation, bursters may change in the forms that, near the center of the spiking oscillation, the trajectories of the whole system move along with the equilibrium states of the fast subsystem, the distance of which is approximated at the difference between amplitude of the self-excitation and the values of the slow variable at the fold bifurcation.