  A mathematical model describing the initial stage of the capture of
oscillatory systems into autoresonance under the action of slowly varying
pumping is considered. Solutions with an infinitely growing amplitude are
associated with the autoresonance phenomenon. Stability of such solutions is of
great importance because only stable solutions correspond to physically
observable motions. We study the stabilizing problem and we show that the
adiabatically varying parametric perturbation with decreasing amplitude in time
can stabilize the unstable autoresonant modes.
