Despite this, it has long been observed that many phenomena in dynamical systems possess a surprising degree of universality. By this, we mean that the important features of a system are independent of the details of that system and are common to all systems in a wide class. The best known example is the so called Feigenbaum period doubling cascade in unimodal maps of the interval. This gives a universal model of the transition to chaos for a wide class of systems. These exponents have been observed in a wide variety of physical experiments ranging from fluid dynamics to electronic circuits.
Universality is thus important from both a practical and a theoretical point of view.
In all known cases today, universality is associated, to a smaller or greater extent, with self-similarity. By far the most powerful mathematical tools for linking the two are so called renormalization operators. Such an operator will act on a given dynamical system to produce an appropriately rescaled version of the original system. If the system is asymptotically self-similar then we can expect to see some limiting behaviour under repeated application of a suitable renormalization operator. In the simplest cases we have one operator and all systems in the given universality class will converge to a fixed point under its action.