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We present a data-driven and interpretable approach for reducing thedimensionality of chaotic systems using spectral submanifolds (SSMs). Emanatingfrom fixed points or periodic orbits, these SSMs are low-dimensional inertialmanifolds containing the chaotic attractor of the underlying high-dimensionalsystem. The reduced dynamics on the SSMs turn out to predict chaotic dynamicsaccurately over a few Lyapunov times and also reproduce long-term statisticalfeatures, such as the largest Lyapunov exponents and probability distributions,of the chaotic attractor. We illustrate this methodology on numerical data setsincluding a delay-embedded Lorenz attractor, a nine-dimensional Lorenz model,and a Duffing oscillator chain. We also demonstrate the predictive power of ourapproach by constructing an SSM-reduced model from unforced trajectories of abuckling beam, and then predicting its periodically forced chaotic responsewithout using data from the forced beam.

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