The analysis of nonlinear phenomena in continuous-time dynamics is one of important topics in the field of engineering, and it has attracted many researchers in recent years. The researchers have tried to solve this problem by simplifying the nonlinear dynamics. We introduce extremely simple oscillators whose dynamics are represented by piecewise-constant equations, and show two examples. One of them is a chaotic spiking oscillator with piecewise-constant vector field. We analytically prove the generation of chaos by using Poincare map which is derived through a simple systematic procedure. Another is a coupled system of piecewise-constant oscillators. The parameter regions of in-phase and anti-phase synchronization are clarified by using a fast calculation algorithm. Some theoretical results are verified in the experimental circuits.