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

2023/09/07 by Ali Tavasoli, Tavasoli, Ali, Heman Shakeri +1 · 2 voices
Computer Science · Mathematics · Physics and Astronomy · #Adaptation and Self-Organizing Systems (nlin.AO) #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Neural Networks and Applications #Optimization and Control (math.OC) #Spectral Theory (math.SP) #cs.LG #math.DS #math.OC #math.SP #nlin.AO

paper · pdf · doi:10.48550/arxiv.2310.12156

openalex publication_date 2023/09/07 · arxiv published 2023/09/07 · arxiv updated 2023/09/07 · openalex created_date 2023/10/21 · openalex updated_date 2026/07/28

Abstract

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.

Discussions

• #arXiv Operator-Based Detecting, Learning, and Stabilizing Unstable Periodic Orbits of Chaotic Attractors arxiv.org/abs/2310.12156 on the use of kernel integral operators within delay coordinates as a [bsky, 3 points, 0 comments]
• # arXiv Operator-Based Detecting, Learning, and Stabilizing Unstable Periodic Orbits of Chaotic Attractors https:// arxiv.org/abs/2310.12156 on the use of kernel integral operators within delay coordi [mastodon, 0 points, 0 comments]

Related

• Deciphering Complexity: Machine Learning Insights into Chaotic Dynamical Systems
• Separation of periodic orbits in the delay embedded space of chaotic attractors
• Data-driven guessing and gluing of unstable periodic orbits
• Nonlinear Dynamics in Complexity Quantification
• Efficient method for detection of periodic orbits in chaotic maps and flows