Supercritical properties of a relaxation oscillator

In this paper we present the analytic expressions of partition lines in the parametric space of a piece-wise smooth mapping describing an electronic r elaxation oscillator. The supercritical regions with permission of period-d oubling bifurcation, prohibition of period-doubling bifurcation, and comple te phase-locking are discussed. Among them the region, where period-doublin g bifurcation is prohibited but chaos is permitted is reported for the firs t time to our knowledge. These kinds of phenomena and regions can be observ ed in a lot of similar systems.