Forward orbit ($O^{+}(x_0)$)

The forward orbit of a state is the collection of states reached as time advances. For a discrete dynamical system $F:X\to X$,For a flow map ${\Phi }_t$, the analogous set is $\{{\Phi }_t(x_0):t\ge 0\}$ over the times for which the solution exists. The initial state is included; its future accumulation points instead form the omega-limit set.

For example, $F(x)=x/2$ with $x_0=1$ gives $1,1/2,1/4,\dots$. Zero is approached but is not one of those iterates, so the forward orbit differs from its closure. Forward evolution also need not determine a unique past: $F(x)=x^2$ sends both $1$ and $-1$ to $1$. The distinction matters when discussing attraction or invariance for a map that cannot be inverted.