Data-Driven Modeling and Forecasting of Chaotic Dynamics on Inertial Manifolds Constructed as Spectral Submanifolds

TL;DR

This paper introduces a data-driven method using spectral submanifolds to reduce the dimensionality of chaotic systems, enabling accurate short-term prediction and statistical reproduction of chaotic attractors.

Contribution

The work develops an interpretable, spectral submanifold-based approach for modeling and forecasting chaos, extending its application to various high-dimensional systems and unforced-to-forced dynamics.

Findings

Accurately predicts chaotic dynamics over a few Lyapunov times.

Reproduces long-term statistical features of chaotic attractors.

Successfully forecasts forced chaotic responses from unforced data.

Abstract

We present a data-driven and interpretable approach for reducing the dimensionality of chaotic systems using spectral submanifolds (SSMs). Emanating from fixed points or periodic orbits, these SSMs are low-dimensional inertial manifolds containing the chaotic attractor of the underlying high-dimensional system. The reduced dynamics on the SSMs turn out to predict chaotic dynamics accurately over a few Lyapunov times and also reproduce long-term statistical features, such as the largest Lyapunov exponents and probability distributions, of the chaotic attractor. We illustrate this methodology on numerical data sets including a delay-embedded Lorenz attractor, a nine-dimensional Lorenz model, and a Duffing oscillator chain. We also demonstrate the predictive power of our approach by constructing an SSM-reduced model from unforced trajectories of a buckling beam, and then predicting its periodically forced chaotic response without using data from the forced beam.