# Operator-Based Detecting, Learning, and Stabilizing Unstable Periodic Orbits of Chaotic Attractors

arXiv: https://arxiv.org/abs/2310.12156 · HTML: https://mlsanity.com/paper/2310.12156/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 37 references that are in the map.
- First posted on arXiv 7 Sep 2023.

## Builds on (most cited first, 6 of 6)

- [From Fourier to Koopman: Spectral Methods for Long-term Time Series Prediction](https://mlsanity.com/paper/2004.00574/) (2020)
- [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017)
- [Residual Dynamic Mode Decomposition: Robust and verified Koopmanism](https://mlsanity.com/paper/2205.09779/) (2022)
- [Kernel methods for detecting coherent structures in dynamical data](https://mlsanity.com/paper/1904.07752/) (2019)
- [Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables](https://mlsanity.com/paper/1808.10551/) (2018)
- [An Interpretable Approach to Load Profile Forecasting in Power Grids using Galerkin-Approximated Koopman Pseudospectra](https://mlsanity.com/paper/2304.07832/) (2023)

## Abstract (arXiv)

This paper examines the use of operator-theoretic approaches to the analysis of chaotic systems through the lens of their unstable periodic orbits (UPOs). Our approach involves three data-driven steps for detecting, identifying, and stabilizing UPOs. We demonstrate the use of kernel integral operators within delay coordinates as an innovative method for UPO detection. For identifying the dynamic behavior associated with each individual UPO, we utilize the Koopman operator to present the dynamics as linear equations in the space of Koopman eigenfunctions. This allows for characterizing the chaotic attractor by investigating its principal dynamical modes across varying UPOs. We extend this methodology into an interpretable machine learning framework aimed at stabilizing strange attractors on their UPOs. To illustrate the efficacy of our approach, we apply it to the Lorenz attractor as a case study.