The capture into parametric autoresonance

TL;DR

This paper explains the capture into parametric autoresonance as a pitchfork bifurcation, linking it to Painleve-2 equations and providing asymptotic connection formulas for the resonance process.

Contribution

It introduces a novel bifurcation perspective on parametric autoresonance and derives asymptotic solutions using Painleve-2 equations.

Findings

Capture into resonance is described as a pitchfork bifurcation.

Asymptotic solutions are connected before and after capture.

Painleve-2 equation characterizes the solution near capture.

Abstract

In this work we show that the capture into parametric resonance may be explained as the pitchfork bifurcation in the primary parametric resonance equation. We prove that the solution close to the moment of the capture is described by the Painleve-2 equation. We obtain the connection formulas for the asymptotic solution of the primary parametric resonance equation before and after the capture using the matching of the asymptotic expansions.