Based upon the combination of the pseudo-arclength continuation method and the Poincaré map defined on a variable return plane, systems of equations are constructed that trace a Takens-Bogdanov bifurcation, a cusp, an isola formation/perturbed bifurcation point and a degenerate period-doubling/secondary Hopf bifurcation of periodic solutions of autonomous ordinary differential equations. The implementation of these ideas into a collection of FORTRAN codes and its application to a five-dimensional system describing an optical bistable system lead to the detection of interesting codimension-3 bifurcations away from the stationary behavior. A winged cusp, a swallow tail, a degenerate hysteresis point, an isola formation point for a codimension-1 loop and two kinds of degenerate Takens-Bogdanov bifurcations of periodic solutions are presented. Finally, based upon the computation of the stability coefficient “a”, attractive tori are found in a systematic way and briefly discussed.