Extending the trapping theorem to provide local stability guarantees for quadratically nonlinear models

The Navier–Stokes equations are partial differential equations to describe the nonlinear convective motion of fluids and they are computationally expensive to simulate because of their high nonlinearity and variables being fully coupled. Reduced-order models (ROMs) are simpler models for evolving the flows by capturing only the dominant behaviors of a system and can be used to design controllers for high-dimensional systems. However it is challenging to guarantee the stability of these models either globally or locally. Ensuring the stability of ROMs can improve the interpretability of the behavior of the dynamics and help develop effective system control strategies. For quadratically nonlinear systems that represent many fluid flows, the Schlegel and Noack trapping theorem [Schlegel and Noack, “On long-term boundedness of Galerkin models,” J. Fluid Mech. 765, 325–352 (2015)] can be used to check if ROMs are globally stable (long-term bounded). This theorem was subsequently incorporated into system identification techniques that determine models directly from data [Kaptanoglu et al., “Promoting global stability in data-driven models of quadratic nonlinear dynamics,” Phys. Rev. Fluids 6, 094401 (2021)]. While the Schlegel and Noack trapping theorem provides global stability criteria for systems with strictly energy-preserving nonlinearities, many physical systems, including those with inflow/outflow boundary conditions, exhibit weakly relaxed energy-preserving structures. This work introduces two key advances: (1) a theorem establishing analytical stability bounds for linear-quadratic systems under relaxed energy-preserving constraints, explicitly quantifying the local stability radius, and (2) the extended trapping SINDy algorithm, which embeds these theoretical guarantees into data-driven system identification. By integrating Lyapunov's direct method with the trapping theorem framework, our approach enables the first provably locally stable models for quadratic dynamics with weakly broken energy-preserving nonlinearities. Several examples are presented to demonstrate the effectiveness and accuracy of the proposed algorithm.