Dynamical Systems

The theory of dynamical systems is a branch of mathematics that has been created in order to describe, explain, and predict the behavior of mathematical models of various phenomena from the experimental sciences. Those models are systems that evolve in time. This evolution is determined by the state of the system at a given moment, and the laws for the evolution, specific for the system. Those laws may be given by a system of differential equations, or just by one transformation that is iterated.

In order to discover and examine possible interesting features of various systems, mathematicians have introduced many systems that are not models of any concrete real phenomena. One can study them by purely theoretical methods or investigate them on computers, or, very often, both. This approach has turned out to be successful. For instance, it has led to the discovery of chaotic phenomena. Those phenomena were subsequently found in most of the experimental sciences and this has had a tremendous impact on the way scientists think. It is now widely recognized that the deterministic nature of a system does not imply long-term validity (the best example is weather prediction), except in a statistical sense.