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

2017/02/06 by Denis Gaidashev, Gaidashev, Denis, Marina Gonchenko +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.DS

paper · pdf · doi:10.48550/arxiv.1702.01651

arxiv created 2017/02/06 · openalex publication_date 2017/02/06 · arxiv updated 2017/02/07 · openalex created_date 2019/05/29 · openalex updated_date 2026/07/28

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 λ₊ is the smaller of the eigenvalues of the saddle point, the angle between the separatrices along the homoclinic orbit satisfies sin α= O(e^-π² \over log |λ₊|)+ O( e^-2 (1-κ) π² \over log |λ₊| ), for any positive κ<1.

Related

• Asymptotic series for the splitting of separatrices near a Hamiltonian bifurcation
• Splitting of separatrices in a family of area-preserving maps that unfolds a fixed point at the resonance of order three
• Period Doubling in Area-Preserving Maps: An Associated One Dimensional Problem
• Reversible perturbations of conservative Hénon-like maps
• Homoclinic bifurcations for the Hénon map