Analysis and synthesis of piecewise-constant oscillators

In this paper, piecewise constant systems whose vector fields are piecewise constant, namely that the righthand side of the state equation is piecewise constant, are considered. Because of piecewise-constant characteristics, the states move proportionally to time except at the moment that vector fields switches, and the trajectories are piecewise-linear in the state space. This systems are extremely simple, however they exhibit some nonlinear phenomena that is due to the nonlinearity of switchings. We are concerned here with nonlinear phenomena of hysteresis oscillators with piecewise-constant dynamics. The typical bifurcation phenomena and chaos of the oscillators are analyzed. We also provide a simple implemented circuit. Using the circuit, typical phenomena are verified in the laboratory.