General oblique projections for model reduction via spectral submanifolds

Slow spectral submanifolds (SSMs) are low-dimensional, attracting, invariant surfaces in the phase space of a dynamical system that carry the dominant nonlinear dynamics. Nearby trajectories rapidly converge to such slow SSMs and synchronize with its internal dynamics thereby enabling mathematically rigorous model reduction to the SSM. In general, oblique projections are required for optimally associating full trajectories off the SSM to their SSM-reduced counterparts. In this work, we establish a rigorous mathematical mapping of the SSM onto its tangent space via general oblique projections and develop a data-driven procedure to efficiently construct SSM-based reduced-order models using these projections. Our approach applies irrespective of the SSM dimension and assumes only limited trajectory information. We illustrate the method on numerical and experimental examples, including nonlinear beam oscillations and artificial muscle actuators.