Those who have knowledge, don't predict. Those who predict, don't have knowledge. —Lao Tzu, 6th-century B.C. Chinese poet
The pendulum we are looking at in Figure 1 obviously has a phase-space portrait (a circle) that fits nicely onto a two-dimensional plane. The reconstructed phase portrait in Figure 5 is also a round curve without intersections. No doubt this is a valid and useful reconstruction. But what if the system in the field has more than two variables, and perhaps their interdependence is not so smooth? While our pendulum simply goes around and around in phase space, there may be other mechanisms with three independent variables, needing three dimensions in their original phase space. Even worse, their portrait may look more like a roller coaster than a flat circle. During the 1980s, scientists worked on this more general problem and found methods and justifications to deal with the situation:

One of the most common mistakes among practitioners of the 1980s was using a fixed tau without thinking about it. If the quality of your data is questionable (because of a bad reconstruction or noise of any origin), it makes no sense to chase for signs of chaos.