Robust computation of higher-dimensional invariant tori from individual trajectories

TL;DR

This paper introduces a practical method for computing higher-dimensional invariant tori in dynamical systems using a single trajectory, employing advanced mathematical tools to achieve robustness and accuracy without prior knowledge.

Contribution

The method uniquely combines reduced rank extrapolation, Bayesian estimation, and lattice basis reduction to compute invariant tori from minimal data without initial guesses or coordinate assumptions.

Findings

Successfully computes 2D invariant tori in standard maps

Demonstrates ability to find 3D invariant tori and islands

Shows robustness and accuracy in practical examples

Abstract

We present a method for computing invariant tori of dimension greater than one. The method uses a single short trajectory of a dynamical system without any continuation or initial guesses. No preferred coordinate system is required, meaning the method is practical for physical systems where the user does not have much \textit{a priori} knowledge. Three main tools are used to obtain the rotation vector of the invariant torus: the reduced rank extrapolation method, Bayesian maximum a posteriori estimation, and a Korkine-Zolatarev lattice basis reduction. The parameterization of the torus is found via a least-squares approach. The robustness of the algorithm is demonstrated by accurately computing many two-dimensional invariant tori of a standard map example. Examples of islands and three-dimensional invariant tori are shown as well.