Analytical study of the structure of chaos near unstable points

TL;DR

This paper investigates the convergence properties of formal integrals near unstable points in 2D Hamiltonian systems, developing methods to analyze chaotic regions analytically through asymptotic curves and their intersections.

Contribution

It explains the convergence differences of formal integrals near stable and unstable orbits and introduces an analytic continuation method for studying chaos near unstable points.

Findings

Convergence radius is infinite in simple mappings, enabling theoretical calculations.

In complex mappings and Hamiltonian systems, the radius is finite, limiting theoretical analysis.

A new analytic continuation method allows studying chaotic regions to arbitrary lengths.

Abstract

In a 2D conservative Hamiltonian system there is a formal integral $\Phi$ besides the energy H. This is not convergent near a stable periodic orbit, but it is convergent near an unstable periodic orbit. We explain this difference and we find the convergence radius along the asymptotic curves. In simple mappings this radius is infinite. This allows the theoretical calculation of the asymptotic curves and their intersections at homoclinic points. However in more complex mappings and in Hamiltonian systems the radius of convergence is in general finite and does not allow the theoretical calculation of any homoclinic point. Then we develop a method similar to analytic continuation, applicable in systems expressed in action-angle variables, that allows the calculation of the asymptotic curves to an arbitrary length. In this way we can study analytically the chaotic regions near the unstable periodic orbit and near its homoclinic points.