Analytical study of chaos and applications

TL;DR

This paper reviews analytical methods for describing chaotic orbits across various systems, including magnetic bottles, maps, and galaxies, highlighting the derivation of homoclinic orbits and convergence domains.

Contribution

It introduces new analytical series and methods for describing chaos and homoclinic structures in dynamical systems, with applications to astrophysics.

Findings

Analytic series (Moser series) describe chaotic orbits around unstable points.

Domains of convergence for these series are identified.

Analytic prolongation methods find homoclinic orbits in Hamiltonian systems.

Abstract

We summarize various cases where chaotic orbits can be described analytically. First we consider the case of a magnetic bottle where we have non-resonant and resonant ordered and chaotic orbits. In the sequence we consider the hyperbolic Henon map, where chaos appears mainly around the origin, which is an unstable periodic orbit. In this case the chaotic orbits around the origin are represented by analytic series (Moser series). We find the domain of convergence of these Moser series and of similar series around other unstable periodic orbits. The asymptotic manifolds from the various unstable periodic orbits intersect at homoclinic and heteroclinic orbits that are given analytically. Then we consider some Hamiltonian systems and we find their homoclinic orbits by using a new method of analytic prolongation. An application of astronomical interest is the domain of convergence of the analytical series that determine the spiral structure of barred-spiral galaxies.