An efficient method for tracing high-resolution invariant manifolds of three-dimensional flows

TL;DR

This paper introduces a new adaptive computational method for accurately and efficiently constructing high-resolution invariant manifolds of unstable periodic orbits in three-dimensional Hamiltonian flows, improving detail and reducing computational cost.

Contribution

The paper presents a novel adaptive algorithm for building invariant manifolds that outperforms random sampling in quality and efficiency, including an approximation to further reduce computations.

Findings

Adaptive method produces high-quality, detailed manifolds.

Compared to random sampling, the method is more efficient and accurate.

The approximation reduces computational cost logarithmically.

Abstract

In Hamiltonian systems subjected to periodic perturbations the stable and unstable manifolds of the unstable periodic orbits provide the dynamical "skeleton" that drives the mixing process and bounds the chaotic regions of the phase space. Determining the behavior of these objects is valuable in physical applications involving asymmetric solenoidal fields or time-dependent Hamiltonian systems. Here we introduce a simple method to calculate an unstable periodic orbit given an initial guess on its position. Then we present an efficient adaptive method to build its high-resolution invariant manifolds to arbitrary length and compare it to a random sampling method with the same computational cost. The adaptive method gives a high-quality representation of the manifolds and reveals fine details that become lost in the random sampling method. Finally, we introduce an approximation to the adaptive method to build the manifolds avoiding redundant calculations and reducing logarithmically the number of computations needed to represent these surfaces.