= Forward orbit
{title2=$O^+(x_0)$}
The <forward orbit> of a state is the collection of states reached as <time> advances. For a <discrete dynamical system> $F:X\to X$,
$$
O^+(x_0)=\{x_0,F(x_0),F^2(x_0),\ldots\}.
$$
For a <flow map> $\Phi_t$, the analogous set is $\{\Phi_t(x_0):t\geq0\}$ 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 $1,1/2,1/4,\ldots$. 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.
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