  A mathematical model of autoresonance in nonlinear systems with combined
parametric and external chirped frequency excitation is considered. Solutions
with a growing amplitude and a bounded phase mismatch are associated with the
autoresonant capture. By applying Lyapunov function method we investigate the
conditions for the existence and stability of autoresonant modes and construct
long-term asymptotics for stable solutions. In particular, we show that
unstable regimes become stable when the system parameters pass through certain
threshold values.
