Editing Dynamical system (section)

===Maps===
A [[Discrete-time dynamical system|discrete-time]], [[Affine transformation|affine]] dynamical system has the form of a [[matrix difference equation]]:
: <math> x_{n+1} =  A x_n + b, </math>
with ''A'' a matrix and ''b'' a vector.  As in the continuous case, the change of coordinates ''x''&nbsp;→&nbsp;''x''&nbsp;+&nbsp;(1&nbsp;−&nbsp;''A'')<sup>&nbsp;–1</sup>''b'' removes the term ''b'' from the equation. In the new [[coordinate system]], the origin is a fixed point of the map and the solutions are of the linear system ''A''<sup>&nbsp;''n''</sup>''x''<sub>0</sub>.
The solutions for the map are no longer curves, but points that hop in the phase space.  The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map.

As in the continuous case, the eigenvalues and eigenvectors of ''A'' determine the structure of phase space.  For example, if ''u''<sub>1</sub> is an eigenvector of ''A'', with a real eigenvalue smaller than one, then the straight lines given by the points along ''α''&nbsp;''u''<sub>1</sub>, with ''α''&nbsp;∈&nbsp;'''R''', is an invariant curve of the map. Points in this straight line run into the fixed point.

There are also many [[List of chaotic maps|other discrete dynamical systems]].