Editing Dynamical systems theory (section)

== Overview ==
Dynamical systems theory and [[chaos theory]] deal with the long-term qualitative behavior of [[dynamical system]]s. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does the long-term behavior of the system depend on its initial condition?"

An important goal is to describe the fixed points, or steady states of a given dynamical system; these are values of the variable that do not change over time. Some of these fixed points are ''attractive'', meaning that if the system starts out in a nearby state, it converges towards the fixed point.

Similarly, one is interested in ''periodic points'', states of the system that repeat after several timesteps. Periodic points can also be attractive. [[Sharkovskii's theorem]] is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system.

Even simple [[nonlinear dynamical system]]s often exhibit seemingly random behavior that has been called ''chaos''.<ref>{{cite journal |last1=Grebogi |first1=C. |last2=Ott |first2=E. |last3=Yorke |first3=J. |year=1987 |title=Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics |journal=[[Science (journal)|Science]] |volume=238 |issue=4827 |pages=632–638 |jstor=1700479 |doi=10.1126/science.238.4827.632 |pmid=17816542 |bibcode=1987Sci...238..632G |s2cid=1586349 }}</ref> The branch of dynamical systems that deals with the clean definition and investigation of chaos is called ''[[chaos theory]]''.