A mathematical model describing the capture of nonlinear systems into the autoresonance by a combined parametric and external periodic slowly varying perturbation is considered. The autoresonance phenomenon is associated with solutions having an unboundedly growing amplitude and a limited phase mismatch. The paper investigates the behavior of such solutions when the parameters of the excitation take bifurcation values. In particular, the stability of different autoresonant modes is analyzed and the asymptotic approximations of autoresonant solutions on asymptotically long time intervals are proposed by a modified averaging method with using the constructed Lyapunov functions.