Operator-based and Data-Driven Methods for Stability Analysis of Nonlinear Dynamical Systems

Nonlinear dynamical systems arise widely in real-world phenomena, including biological systems, collective behavior, and climate systems. Understanding the behavior of these nonlinear phenomena and revealing their underlying dynamical structures and physical meanings are important challenges in information mathematics. In particular, the modeling and analysis of stability and dynamics in nonlinear dynamical systems are closely related to problems in control, optimization, and prediction.

In recent years, data-driven approaches have emerged as a promising direction for understanding nonlinear dynamical systems. In particular, Koopman operator theory provides a new perspective by describing the evolution of observables through a linear operator, offering an operator-theoretic framework for analyzing the structure and dynamics of nonlinear systems directly from data. Motivated by these developments, my research focuses on developing data-driven algorithms to extract essential dynamical structures from complex physical phenomena through operator-based spectral analysis, dimensionality reduction, and integration with machine learning and optimization. Furthermore, we aim to apply these operator-based and data-driven methods to practical systems, addressing nonlinear phenomena that are difficult to capture with conventional approaches and exploring their potential in mathematical and physical engineering.