Creation of Homoclinic Tangencies in Hamiltonians by the Suspension of Poincar\'e Sections

TL;DR

This paper demonstrates that for any Hamiltonian on a symplectic 4-manifold, one can find a nearby Hamiltonian whose energy surface exhibits either Anosov dynamics or contains a homoclinic tangency, using Hamiltonian suspensions.

Contribution

It introduces a method to create homoclinic tangencies in Hamiltonian systems via suspension of Poincaré sections, expanding control over Hamiltonian dynamics.

Findings

Existence of Hamiltonians with homoclinic tangencies near any given Hamiltonian.

Construction of Hamiltonian suspensions for symplectomorphisms.

Regular energy surfaces can be made Anosov or contain tangencies through small perturbations.

Abstract

In this note we show that for any Hamiltonian defined on a symplectic 4-manifold M and any point p in M, there exists a C2-close Hamiltonian whose regular energy surface through p is either Anosov or it contains a homoclinic tangency. Our result is based on a general construction of Hamiltonian suspensions for given symplectomorphisms on Poincar\'e sections already known to yield similar properties.