Asymptotic behaviour of measure for captured trajectories into parametric autoresonance

We study an asymptotic behaviour of parametric autoresonance for nonlinear equation. Main result of this work is statement about asymptotic behaviour of measure for captured trajectories. We show that the measure is bounded for infinite time interval. To find this, we obtain an asymptotic expansion for capture and asymptotic behaviour of split separatrices for intermediate small amplitudes. In this study, we have used two scaling method for studying the interval of capture. We reduce the problem to equation which defines the separatrix splitting and apply Melnikov’s theory to define the splitting of the separatrices. The obtained result shows important difference between parametric and nonparametric autoresonance, because the measure for captured trajectories for nonparametric autoresonance is not bounded.