  The influence of multiplicative white noise on the resonance capture of
strongly nonlinear oscillatory systems under chirped-frequency excitations is
investigated. It is assumed that the intensity of the perturbation decays
polynomially with time, and its frequency grows according to a power low.
Resonant solutions with a growing amplitude and phase, synchronized with the
excitation, are considered. The persistence of such a regime in the presence of
stochastic perturbations is discussed. In particular, conditions are described
that guarantee the stochastic stability of the resonant modes on infinite or
asymptotically large time intervals. The technique used is based on a
combination of the averaging method, stability analysis and construction of
stochastic Lyapunov functions. The proposed theory is applied to the Duffing
oscillator with a chirped-frequency excitation and noise.
