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

Abstract

We consider the conservative Hénon 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 $$\sin \alpha =O(e^{-\frac{{\pi }^2}{\log |{\lambda }_{+}|}})+O\left(e^{-2(1-\kappa )\frac{{\pi }^2}{\log |{\lambda }_{+}|}}\right),$$ for any positive $\kappa <1$.