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.

References

Local models offer insight near fixed points but fail to capture global nonlinear behavior, which remains a major open problem [brunton2019data].

— Convergent Methods for Koopman Operators on Reproducing Kernel Hilbert Spaces  (2506.15782 - Boullé et al., 18 Jun 2025) in Section 1 (Introduction)

Similarly, there is still the open question regarding the complex behavior of the nonlinear evolution of KBMs when considering the global approach.

— Electromagnetic pressure-gradient-driven instabilities with moderate high mode numbers in tokamak plasmas  (2608.28367 - Narbutt et al., 28 Aug 2026) in Summary

How the system evolves and how a continued attack may ensue thereafter is a subject of future work.

— Targeted Power System Frequency Attack via the Selection of Maliciously Controlled Inverters  (2608.28533 - Zou et al., 28 Aug 2026) in Section 5.2, Example Attack

Beyond the results developed here, this framework poses a constructive question: can the locally optimal response coordinates identified by K_O be combined into a stable nonlinear filter that remains accurate across measurement records? Answering this would connect the local dimension--error bound to a globally effective observable filter.

— Realization Theory for Quantum Filtering: Classifying Continuously Monitored Bosonic Systems  (2609.09663 - Emerson, 9 Sep 2026) in Introduction, final paragraph before Section 2

This approach raises the question of whether all nonlinear solution structures can be correctly represented.

— Fully Discrete Active Flux Methods for the Euler Equations: Exploring Different Reconstructions and New Multi-Dimensional Evolution Operators  (2609.34669 - Helzel et al., 28 Sep 2026) in Section 5, Approximation of Transonic Rarefaction Waves

Overall, identifying data-driven nonlinear IBR models able to fully capture the interplay among static, small- and large-signal phenomena as discussed in Section \ref{sec:model-based} remains still an open issue.

— Stability Analysis in IBR-Dominated Grids: Model-Based and Data-Driven Methodologies  (2610.22227 - Cecati et al., 15 Oct 2026) in Section 3, subsection “Black-box nonlinear model of IBRs for large-signal analyses”