Width of homoclinic zone for quadratic maps

TL;DR

This paper investigates the width of the chaotic zone near a Bogdanov-Takens bifurcation in quadratic maps, providing numerical tests of an asymptotic expansion for this zone's size across different families.

Contribution

It offers a numerical validation of an asymptotic expansion for the width of the homoclinic chaotic zone in quadratic maps near bifurcation points.

Findings

The chaotic zone width can be accurately predicted by the asymptotic expansion.

The zone's width is exponentially small and varies across different families.

Numerical results confirm the theoretical asymptotic predictions.

Abstract

We study several families of planar quadratic diffeomorphisms near a Bogdanov-Takens bifurcation. For each family, the associate bifurcation diagram can be deduced from the interpolating flow. However, a zone of chaos confined between two lines of homoclinic bifurcation that are exponentially close to one-another is observed. The goal of this paper is to test numerically an accurate asymptotic expansion for the width of this chaotic zone for different families.