The oscillation and bifurcation of a switching system composed of jump circuits

The complex dynamical evolution of a circuit system composed of two nonlinear circuit subsystems, which is switched by a periodic switching, is investigated. According to the fact that the magnification of an open-loop operational amplifier is maximum magnification, namely, the operational amplifier is always in a positive or negative saturated state, when an input voltage becomes positive from negative through zero, the output voltage jumps from the positive saturation into negative saturation. In this paper the jump function is selected as a nonlinear part in subsystems. Firstly through the stability analysis of the subsystems, their oscillation behaviors in the parameter space are given correspondingly. Secondly the complex oscillation behavior and mechanism of the switched system are discussed in the parameter space of one subsystem. The periodic orbit of the switched system is divided into four parts, influenced by non-smooth characteristics of the subsystems and switching. With the variation of the parameters, grazing bifurcation appears, and then the whole periodic orbit is separated into two symmetrical periodic oscillations. Finally the convesion of switching points into the periodic oscillation is given,and the mechanism at switching point is discussed.