  For the ordinary differential equation (ODE) $\dot{x}(t) = f(t,x)$, $x(0) =
x_0$, $t\geq 0$, $x\in R^d$, assume $f$ to be at least continuous in $t$ and
locally Lipshitz in $x$, and if necessary, several times continuously
differentiable in $t$ and $x$. We associate a conditioning function $E(t)$ with
each solution $x(t)$ which captures the accumulation of global error in a
numerical approximation in the following sense: if $\tilde{x}(t;h)$ is an
approximation derived from a single step method of time step $h$ and order $r$
then $\norm{\tilde{x}(t;h) - x(t)} < K(E(t)+\epsilon)h^r$ for $0\leq t\leq T$,
any $\epsilon > 0$, sufficiently small $h$, and a constant $K>0$.
  Using techniques from the stability theory of differential equations, this
paper gives conditions on $x(t)$ for $E(t)$ to be upper bounded linearly or by
a constant for $t\geq 0$. More concretely, these techniques give constant or
linear bounds on $E(t)$ when $x(t)$ is a trajectory of a dynamical system which
falls into a stable, hyperbolic fixed point; or into a stable, hyperbolic
cycle; or into a normally hyperbolic and contracting manifold with
quasiperiodic flow on the manifold.
