Analytical invariant manifolds near unstable points and the structure of chaos

TL;DR

This paper investigates the convergence of series representations of invariant manifolds near unstable points in Hamiltonian systems, introducing a new method to compute these manifolds over large extents for analyzing homoclinic chaos.

Contribution

It presents a novel series composition method using action-angle variables, enabling long-range analytical computation of invariant manifolds in Hamiltonian systems.

Findings

Series convergence does not reach homoclinic points in Hamiltonian systems.

Higher series order and numerical precision improve approximation accuracy.

New method allows analytical study of homoclinic chaos over extended regions.

Abstract

It is known that the asymptotic invariant manifolds around an unstable periodic orbit in conservative systems can be represented by convergent series (Cherry 1926, Moser 1956, 1958, Giorgilli 2001). The unstable and stable manifolds intersect at an infinity of homoclinic points, generating a complicated homoclinic tangle. In the case of simple mappings it was found (Da Silva Ritter et al. 1987) that the domain of convergence of the formal series extends to infinity along the invariant manifolds. This allows in practice to study the homoclinic tangle using only series. However in the case of Hamiltonian systems, or mappings with a finite analyticity domain,the convergence of the series along the asymptotic manifolds is also finite. Here, we provide numerical indications that the convergence does not reach any homoclinic points. We discuss in detail the convergence problem in various cases and we find the degree of approximation of the analytical invariant manifolds to the real (numerical) manifolds as i) the order of truncation of the series increases, and ii) we use higher numerical precision in computing the coefficients of the series. Then we introduce a new method of series composition, by using action-angle variables, that allows the calculation of the asymptotic manifolds up to an a arbitrarily large extent. This is the first case of an analytic development that allows the computation of the invariant manifolds and their intersections in a Hamiltonian system for an extent long enough to allow the study of homoclinic chaos by analytical means.