Data-driven reduced modeling of delayed dynamical systems via spectral submanifolds

We show how the recent extension of spectral submanifold (SSM) theory to delay differential equations (DDEs) enables data-driven model reduction of nonlinear delay systems. First, using a scalar DDE with a single discrete delay, we compare equation-based and data-driven SSM reductions, to illustrate the need for the latter. We then use the same algorithm to obtain purely data-driven, SSM-reduced, delay-free ordinary differential equation models for several nonlinear delayed systems. Our approach requires no information about the form of the underlying DDE, or about the number and magnitude of the delays it contains. Our SSM-reduced, low-dimensional models remain predictive even for chaotic dynamics. We also illustrate the use of parametric SSM reduction to capture bifurcations in systems with both distributed and discrete delays. Finally, we extend the theoretical underpinning of delayed SSM reductions to non-autonomous systems with periodic delays and apply these results to experimental data from a control system with feedback delay and quantization.