An asymptotic structure of the bifurcation boundary of the perturbed Painlev\'e-2 equation

TL;DR

This paper investigates the complex spiral structure of the bifurcation boundary in the perturbed Painlevé-2 equation, providing analytical and numerical insights into how perturbations influence the boundary's asymptotic behavior.

Contribution

It derives equations for the modulation of the bifurcation boundary under perturbations and characterizes its asymptotic spiral structure.

Findings

Bifurcation boundary exhibits a spiral structure.

Analytical equations for boundary modulation are obtained.

Numerical results support the analytical findings.

Abstract

Solutions of the perturbed Painlev\'e-2 equation are typical for describing a dynamic bifurcation of soft loss of stability. The bifurcation boundary separates solutions of different types before bifurcation and before loss of stability. This border has a spiral structure. The equations of modulation of the bifurcation boundary depending on the perturbation are obtained. Both analytical and numerical results are given