Passage through resonance for a system with time-varying parameters possessing a single trapped mode

We consider a forced oscillations of an infinite-length mechanical system, with time-varying parameters, possessing a single trapped mode characterized by frequency $\Omega_0(\epsilon t)$. The system is a string, lying on the Winkler foundation, and equipped with a discrete linear mass-spring oscillator of time-varying stiffness. The discrete oscillator is subjected to harmonic external force with constant frequency $\hat\Omega$. In the case of the passage through the resonance, we obtain the principal term of the asymptotic expansion describing the motion of the inclusion (e.g. the mass-spring oscillator). To do this we use the combination of two asymptotic approaches. The first one was suggested in Gavrilov and Indeitsev (2002) and used in our recent study Gavrilov et al. (2019b) to describe the free localized oscillation in the system under consideration. The second one was used in Kevorkian (1971, 1974) to describe the passage through the resonance in a single degree of freedom system. The obtained result was verified by independent numerical calculations based on solution of the corresponding PDE by means of the method of finite differences. The comparison demonstrates a good mutual agreement.