Capture and holding of resonance far from equilibrium

Capture into resonance occurs in nonlinear oscillating systems. The study of mathematical models of this phenomenon is a part of a modern theory of nonlinear oscillations. The known result in this field were obtained by averaging method in the asymptotic regime with a small parameter. In this way, an initial stage of the capture into resonance was studied in details.
The matter of this approach is an asymptotic passage to a model equation of mathematical pendulum kind. In the present work we consider an asymptotic construction at long time, which describes a slow evolution of a solution captured into resonance. The main aim is to determine a time interval, during which the resonance is held. The problem is reduced to studying a perturbation of a model equation of pendulum type. Our main success is the description of the time interval, in which the resonance is captured, and the description is given in terms of the data in the original problem. Formally we consider a nonlinear oscillating system with a small perturbation. The perturbation is described by an external pumping with a prescribed slowly changing frequency. For the solutions captured into the resonance, we consider asymptotics with respect to the small parameter. We write out an equation, the solution to which allows us to find the time of the capturing into resonance.