Global nonlinear behavior beyond local models

Develop approaches that accurately capture the global nonlinear behavior of dynamical systems beyond local models near fixed points, addressing the inability of local linearizations to represent global dynamics in complex systems.

Background

The paper highlights foundational challenges in analyzing nonlinear dynamical systems, particularly the limitations of local models which provide insight only near fixed points. These local approaches fail to capture the global, often highly nonlinear behavior exhibited by many real-world systems across fields such as fluid dynamics, neuroscience, and climate science.

This open problem motivates operator-theoretic frameworks, including Koopman operator methods, which aim to provide a form of global linearization by acting on observables rather than state-space trajectories. The authors present convergent algorithms for Koopman spectral analysis on RKHSs, thereby addressing related computational barriers; however, the broader question of developing methods that capture global nonlinear behavior—beyond local neighborhoods—remains open in the field.