Data-driven modeling of subharmonic forced response due to nonlinear resonance.

Complex behavior in nonlinear dynamical systems often arises from resonances, which enable intricate energy transfer mechanisms among modes that otherwise would not interact. Theoretical, numerical and experimental methods are available to study such behavior when the resonance arises among modes of the linearized system. Much less understood are, however, resonances arising from nonlinear modal interactions, which cannot be detected from a classical linear analysis.

Data-driven modeling and forecasting of chaotic dynamics on inertial manifolds constructed as spectral submanifolds.

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.

Nonlinear model reduction to fractional and mixed-mode spectral submanifolds.

A primary spectral submanifold (SSM) is the unique smoothest nonlinear continuation of a nonresonant spectral subspace E of a dynamical system linearized at a fixed point. Passing from the full nonlinear dynamics to the flow on an attracting primary SSM provides a mathematically precise reduction of the full system dynamics to a very low-dimensional, smooth model in polynomial form. A limitation of this model reduction approach has been, however, that the spectral subspace yielding the SSM must be spanned by eigenvectors of the same stability type.

Data-driven modeling and prediction of non-linearizable dynamics via spectral submanifolds.

We develop a methodology to construct low-dimensional predictive models from data sets representing essentially nonlinear (or non-linearizable) dynamical systems with a hyperbolic linear part that are subject to external forcing with finitely many frequencies. Our data-driven, sparse, nonlinear models are obtained as extended normal forms of the reduced dynamics on low-dimensional, attracting spectral submanifolds (SSMs) of the dynamical system. We illustrate the power of data-driven SSM reduction on high-dimensional numerical data sets and experimental measurements involving beam oscillations, vortex shedding and sloshing in a water tank.