Operator-Based Detecting, Learning, and Stabilizing Unstable Periodic Orbits of Chaotic Attractors
This work addresses the challenge of controlling chaotic systems for applications in physics and engineering, presenting an incremental improvement through a novel interpretable machine learning framework.
The paper tackles the problem of analyzing chaotic systems by detecting, identifying, and stabilizing unstable periodic orbits (UPOs) using operator-theoretic methods, demonstrating its efficacy on the Lorenz attractor as a case study.
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.