 Discrete dynamical system

A discrete dynamical system specifies a phase space $X$ and a map $F:X\to X$. An initial state $x_0$ determines its forward orbit byThe phase space may be an interval, a plane, a manifold or another suitable set; one-dimensional iteration is one special case. A two-sided orbit additionally needs backward states, which are uniquely determined when $F$ is invertible.

A fixed point obeys $F(x^{\ast })=x^{\ast }$. A point of least period $k$ obeys $F^k(x)=x$ with no smaller positive return time; its orbit is a periodic orbit. For example, $F(x)=x/2$ on the real line gives $x_n=2^{-n}{x}_0$, converging to the attracting fixed point zero. An irrational circle rotation instead gives aperiodic orbits; discrete time alone does not imply chaos.