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

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.

Figures (17)

• Figure 1: Reconstruction and prediction results of Lorenz attractor in 3D. The inertial manifold is reconstructed by observing only $x$ coordinate as a scalar time serie, delay embedded into 7D, and performing model reduction to 3D. Dynamics on SSM is represented by a fifth-order polynomial.
• Figure 2: Time series forecasting results for delay-embedded Lorenz attractor, and its reconstructed Lyapunov exponent. Subplots (\ref{['fig:3D_pre']}) and (\ref{['fig:3D_NMTE']}) show more accurate predictions from SSM-reduction than from SINDy over approximately 2.97 Lyapunov times (NMTE less than 0.1). Subplot (\ref{['fig:3D_le']}) plots the separation of nearby trajectories against time. The red prediction made by the SINDy model has the maximal Lyapunov exponent 0.23 compared with the real MLE 0.90 of system (\ref{['lorenz_eqa']}). A blue linear fit to the SSM prediction confirms that trajectories separate exponentially, and the system is chaotic with the maximal Lyapunov exponent 0.89. At the same time, the red prediction made by SINDy suggests that the system is not chaotic.
• Figure 3: The chaotic attractor of the 9D Lorenz model (\ref{['9Dlorenz_eqa']}) in the 7D delay-embedded space, projected onto the coordinates $y_1$, $y_3$ and $y_7$.
• Figure 4: Trajectory prediction in the time domain (plot (\ref{['fig:9D_pre']})) and in the delay-embedding space (plot (\ref{['fig:9D_pre2']})) from the reduced models for an initial condition not contained in the training data. The length of the SSM prediction (blue) interval is approximately 5.94 Lyapunov times, while the SINDy model (red) fails for trajectory integration at around 1.82 Lyapunov times. We obtain the prediction of the SSM-reduced model by making predictions on 3D reduced coordinates using the kNN method, then projecting the trajectory back in 7D delay-embedded space. The training set contains $1.5$ million points.
• Figure 5: Trajectory prediction error (\ref{['fig:9D_NMTE']}) and the Lyapunov exponent analysis (\ref{['fig:9D_le']}) on test data of the 3D SSM-reduced model. We compute the time average of the NMTE and the nearby trajectory separation rate over 200 test trajectories, but only plot 20 of them in this figure. The average trajectory separation rate of system (\ref{['9Dlorenz_eqa']}) is 0.032, which is closely approximated by the mean Lyapunov exponent 0.033 computed from subplot (\ref{['fig:9D_le']}) of the SSM-reduced model.