非线性电路的簇发现象及分岔机制

通过在立方非线性Hartley模型中引入交变的电流源，并选定适当的参数和激励频率，建立了具有快慢行为的两时间尺度周期激励电路系统．由Hopf分岔的产生条件，推导了对应自治系统Hopf分岔的第一Lyapunov系数解析表达式，并在数值计算中得到了验证．结合该系数，重点分析了系统中的快慢行为，给出了典型的周期簇发现象及其相应的分岔模式，并利用自治系统和转换相图从分岔角度指出了该种簇发现象的产生机理．

Bursting phenomena as well as bifurcation mechanism in nonlinear circuit

By introducing time-dependent current source to Hartley model with cubic nonlinearity and choosing suitable values of parameter and excited frequency, we produce a periodically excited fast-slow electric circuit with two time-scales. The condition for the occurrence of Hopf bifurcation is used to derive the analytical expression of the first Lyapunov coefficient, which is validated by numerical simulation. The coefficient, as well as the bifurcation theory is employed to investigate the fast-slow effect in the system, which leads to the typical periodic bursting in the associated bifurcation modes. Based on the autonomous system and the transformed phase-portraits, the mechanism of the bursting phenomenon is presented from the standpoint of bifurcation.