Painlevé II Equation As a Model of a Resonant Interaction of Oscillators

Abstract

We consider a system of differential equations that describes the interaction of two weakly coupled nonlinear oscillators. The initial data are such that, in the absence of coupling, the first oscillator is far from equilibrium, the second oscillator is near equilibrium, and their natural frequencies are close to each other. We study the resonance capture phenomenon, when the frequencies of the coupled oscillators remain close and the amplitudes of their oscillations undergo significant time variations; in particular, the second oscillator moves far away from the equilibrium. We find that the initial stage of the resonance capture is described by a solution of the second Painlevé equation. The description is obtained under an asymptotic approximation with respect to a small parameter corresponding to the coupling factor.