This study deals with a forced self-excited vibrating system accompanied by dry friction as an example of a nonlinear system with discontinuities. The numerical method used here is the direct numerical integral method as we have presented previously, which is a shooting method able to obtain highly accurate periodic solutions of systems with discontinuities. The resonance curves of entrained harmonic, higher-harmonic, and subharmonic vibrations are obtained. Chaos is also found. The influences of amplitude and frequency of an external force on the resonance curves and the stability of solutions are discussed. It is also found that bifurcations are realized from the discontinuous characteristics of fractional force. Due to this type of bifurcation, periodic solution changes its stability abruptly and chaos occurs suddenly halfway on the period doubling route in the system. This characteristic of periodic solution is verified using characteristic multipliers, and that of chaos is verified using Lyapunov exponents, bifurcation diagrams, and Poincaré maps. As another system with discontinuity, preloaded compliance system is dealt with, and a similar discontinuity of stability and route to chaos are briefly shown.