Physics

Nonlinear system

Invariant (physics)

Forcing (mathematics)

Mathematics

Mathematical physics

Dynamical systems theory

Optics

Data mining

Reduction (mathematics)

Data reduction

Mathematical analysis

Artificial intelligence

Vibration

Mechanical system

Amplitude

Control (management)

Geometry

Computer science

Quantum mechanics

Applied mathematics

Nonlinear dynamical systems

Control theory (sociology)

Linear system

LTI system theory

Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

While data-driven model reduction techniques are well-established for linearizable mechanical systems, general approaches to reducing nonlinearizable systems with multiple coexisting steady states have been unavailable. In this paper, we review such a data-driven nonlinear model reduction methodology based on spectral submanifolds. As input, this approach takes observations of unforced nonlinear oscillations to construct normal forms of the dynamics reduced to very low-dimensional invariant manifolds. These normal forms capture amplitude-dependent properties and are accurate enough to provide predictions for nonlinearizable system response under the additions of external forcing. We illustrate these results on examples from structural vibrations, featuring both synthetic and experimental data. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

Data-driven nonlinear model reduction to spectral submanifolds in mechanical systems