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.