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arXiv · 2505.08715

Robust computation of higher-dimensional invariant tori from individual trajectories

Maximilian Ruth·Jackson Kulik·Joshua Burby
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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.

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Maximilian Ruth, Jackson Kulik, Joshua Burby. 2025-05-13. Robust computation of higher-dimensional invariant tori from individual trajectories. https://arxiv.org/abs/2505.08715

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