Tools to detect structures in dynamical systems using Jet Transport

Abstract

This paper is devoted to the development of some dynamical indicators that allow the determination of regions and structures that separate different dynamic regimes in autonomous and non-autonomous dynamical systems. The underlying idea is closely related to the Lagrangian Coherent Structures concept introduced by Haller. In the present paper, instead of using the Cauchy-Green tensor, that determines the domains where the flow associated to a differential equation is expanding in the normal direction, the Jet Transport methodology is used. This is a semi-numerical tool, that has as basic ingredients a polynomial algebra package and a numerical integration method, allowing, at each integration step, the propagation under a flow of a neighbourhood $U$ instead of a single initial condition. The output of the procedure is a polynomial in several variables that represents the image of $U$ up to a selected order, containing high order terms of the variational equations. Using these high order representation, the places where the normal direction expands can be easily detected, in a similar manner as the procedures for calculating the Lagrangian Coherent Structures do. In order to illustrate the methodology, first the results obtained in the determination of the separatrices of the simple and the periodically perturbed pendulum are given. Later, the applications to the circular restricted three body problem are considered, where the aim is the detection of invariant manifolds of libration point orbits, as well as in the non-autonomous vector field defined by the elliptic restricted three body problem.