Thanks. What I don’t understand is how searching for previous patterns that are similar helps in predicting timelines that are chaotic (it seems to be quite good at that).

It only helps because the chaotic system under consideration has periodic components.

The attractor shown in figure 1e has such periodic components, and identifying these does help, but only with very near term forecasting. When the accumulated forecast error crosses a threshold, it suddenly causes a large phase error, best seen from about point 75 onwards in the x and y components. From that point onwards the forecast is useless.