Time-Scale Separation from Diffusion-Mapped Delay Coordinates

It has long been known that the method of time-delay embedding can be used to reconstruct nonlinear dynamics from time series data. A less-appreciated fact is that the induced geometry of time-delay coordinates increasingly biases the reconstruction toward the stable directions as delays are added. This bias can be exploited, using the diffusion maps approach to dimension reduction, to extract dynamics on desired time scales from high-dimensional observed data. We demonstrate the technique on a wide range of examples, including data generated by a model of meandering spiral waves and video recordings of a liquid-crystal experiment.