# Exponentially small splitting of separatrices near a period-doubling bifurcation in area-preserving maps

## TL;DR

This paper proves that in the area-preserving Hénon map near a period-doubling bifurcation, the splitting of separatrices occurs exponentially small, with a precise asymptotic estimate involving eigenvalues.

## Contribution

It provides a rigorous asymptotic estimate for the exponentially small splitting of separatrices near a period-doubling bifurcation in the conservative Hénon family.

## Key findings

- Separatrices split exponentially close to the bifurcation point.
- The angle between separatrices is asymptotically proportional to an exponential function of eigenvalues.
- The splitting magnitude is quantified with explicit exponential bounds.

## Abstract

We consider the conservative H\'enon family at the period-doubling bifurcation of its fixed point and demonstrate that the separatrices of the fixed saddle point nearing the bifurcation split exponentially: given that $\lambda_+$ is the smaller of the eigenvalues of the saddle point, the angle between the separatrices along the homoclinic orbit satisfies $$\sin \alpha = O(e^{-{\pi^2 \over \log |\lambda_+|}})+ O\left( e^{-2 (1-\kappa) {\pi^2 \over \log |\lambda_+|}} \right),$$ for any positive $\kappa<1$.