Stochastic self-oscillator is one of the most intriguing features in nonlinear dynamics, and usually can be observed at the accumulation point of successive period doubling bifurcation. In regard to this problem, the substance deals with the stability and bifurcation aspects of oscillatory motions, fine structure of trajectories, typical properties shown in the flows and maps, and the mathematical rigor in measuring the stochasticity concerned with applications to science and engineering problems.