  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 behaviour of such solutions when the parameters of
the excitation pass through 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.
