Based on the operation principle and state equations of multi-frequency (MF) controlled switching converter, the discrete iterative map model is established, the corresponding characteristics equation and Lyapunov exponent of MF controlled switching converter operating in discontinuous conduction mode are derived. According to the discrete iterative map model, its bifurcation behaviors are investigated. It is found that the border-collision bifurcation and the multi-period behaviors occur in converter, when the circuit parameters are varied. When the voltage error boundary is too large, the converter behaves as self-similarity phenomenon, and when the voltage error boundary is too small, the converter behaves as frequency-mixing phenomenon. The converter stability is investigated by Lyapunov exponent, which validates the correctness of bifurcation analysis. Time-domain waveforms and phase portraits of converter with the load varied, are analyzed by circuit simulation, which validates the validity of discrete iterative map model and the correctness of theoretical analysis. Finally, simulation results in this paper are verified by experimental results.