Width of the Homoclinic Zone in the Parameter Space for Quadratic Maps

Abstract

We study several families of planar quadratic diffeomorphisms near a Bogdanov–Takens bifurcation. For each family, the skeleton of the associated 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.

Keywords:

• Primary 37D45, 37E30, 37G10
• Bogdanov map
• Bogdanov-Takens bifurcation
• Hénon map
• separatic splitting
• splitting function