Abstract: System modeling and local stability analysis of equilibrium points are indispensable for analysis of bifurcation and chaos behaviors observed in piecewise autonomous oscillation system. This paper presents a novel dynamic discrete mapping modeling method and local stability criterion of equilibrium points. This modeling method sets up discrete mapping model and boundary equations in analytic form respectively. With the aid of Newton-Raphson algorithm, the boundary equations can be solved dynamically. Combining the mapping model with boundary dynamic solutions, this mapping model can fulfill the requirements of fast-scale bifurcation analysis. To verify this modeling method, an autonomous oscillation circuit used in power electronics soft switched converter is constructed. A frequency bifurcation phenomenon is captured in both simulation results and experiment system. The analysis results of this bifurcation phenomena show consistency between model simulation and experiment system.