Resonance Capture in a System of Two Oscillators Near Equilibrium

We consider a system of differential equations describing the interaction of two weakly coupled nonlinear oscillators. We assume that one oscillator is initially far from equilibrium, the other is near equilibrium, and their frequencies are close. We study the effect of resonance capture, when the frequencies of the coupled oscillators remain close while the oscillation energies change in time significantly; in particular, the second oscillator goes far from equilibrium. We find that the initial stage of resonance capture is described by the second Painlev´e equation. We obtain such a description in the asymptotic approximation in a small parameter corresponding to the coupling constant.