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

TL;DR

This work addresses stability guarantees for reduced-order models of quadratically nonlinear dynamical systems, notably those arising from Navier–Stokes discretizations. It extends the Schlegel–Noack trapping framework to local stability by relaxing energy-preserving constraints and integrates this into a data-driven identification approach via an extended trapping SINDy algorithm that leverages a Lyapunov matrix to bound the stability radius. The method yields a-priori locally stable data-driven models and provides explicit radii $\rho_-$ and $\rho_+$ that quantify the dynamically stable region around the origin, demonstrated on noisy Lorenz, dysts chaotic systems, Von Karman vortex streets, and lid-driven cavity flows. This approach enhances robustness and interpretability of ROMs and offers a principled path to stability-aware model discovery for fluid and plasma dynamics with open-boundary or weakly dissipative effects.

Abstract

The Navier Stokes equations (NSEs) are partial differential equations (PDEs) 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 (JFM, 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. In this work, we relax the quadratically energy-preserving constraints in this theorem, and then promote local stability in data-driven models of quadratically nonlinear dynamics. First, we prove a theorem outlining sufficient conditions to ensure local stability in linear-quadratic systems and provide an estimate of the stability radius. Second, we incorporate this theorem into system identification methods and produce a-priori locally stable data-driven models. Several examples are presented to demonstrate the effectiveness and accuracy of the proposed algorithm.

Figures (9)

• Figure 1: Sketch of an attracting trapping region. $\bm \Gamma_1$ (in red, solid) and $\bm \Gamma_2$ (in blue, solid) are two trajectories starting outside an attracting trapping region $\bm{\mathcal{N}}$ where all trajectories will eventually enter in and stay there forever.
• Figure 2: Sketch of the trapping region. $\dot v < 0$ is observed in $\bm \Omega_{\rho_+}\setminus\bm \Omega_{\rho_-}$ and long-term stability is guaranteed for any initial conditions $\|\bm y(t_0)\|_{\bm P} =\|\bm z(t_0)\|_2 \leq \rho_+$.
• Figure 3: An example trajectory (dashed blue) beginning outside of the locally attracting trapping region $\bm B_{\rho_-}$ and remaining in once it enters in it.
• Figure 4: The progress of stability radius growth during optimization of identifying Lorenz attractor from data. The green area below the dashed line is the smallest ball-shape trapping region found by the model and the blue area between the solid line and the dashed line corresponds to the dotted area in Fig. \ref{['fig:trajectory_sketch']}, where $\dot K < 0$. All trajectories in this area will eventually fall into the green one. The red area outside means no guarantees for the stability and trajectories may be long-time unstable.
• Figure 5: A collection of Lorenz attractors with additive Gaussion noise. Left: Lorenz trajectories with noise for training (black), the corresponding extended trapping SINDy models (red) and the corresponding base SINDy models (green). Right: Lorenz trajectories without noise for test (black, their prediction by extended trapping SINDy models (red) and prediction by base SINDy models (green).

Theorems & Definitions (7)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1
• Theorem 2
• proof