There is a fundamental difference between a discrete map and a discrete
solution to an ODE. Local errors of the former are restricted to being
``space like'' -- there is no concept of the passage of time between
iterations of the map. The latter, however, can have errors in space
*as well as time*. That is, the numerical error in the length of
each timestep can accumulate, leading the numerical solution to have a
slightly different time scale than the real system.

In the integration of periodic or almost periodic systems, like the solar system, this is also known as phase error, because the numerical solution may have a slightly different period than the true solution. Thus, although the ``orbit'' may be reproduced correctly by the numerical solution, the time at which the real particle and simulated particle pass a certain point may differ. This is the case even if the integrator is symplectic [76, 26].

Thus, when attempting to shadow a numerical solution, it may be necessary to ``rescale'' time [16, 17, 18, 80]. The definition of a shadow of an ODE system must then be modified:

*Definition of ODE shadowing*: An approximate trajectory
with timesteps is -shadowed by a true solution
if there exists a sequence of points with
timesteps such that
where is the -flow of the system,
and and .

In other words, the numerical solution is shadowed if it closely
follows the *path* of a true solution, but at time *t* it
is allowed to be up to time units ahead of, or behind, the true
solution. This linear growth of errors is due to a
lack of hyperbolicity in the direction of the flow in phase
space [80]. For large *t*, this can be a significant difference,
so a shadowing algorithm which does not take the re-scaling of time
into account is likely to grossly underestimate the length of the shadow.
Coomes, Koçak, and Palmer
[17, 18] dramatically demonstrate
this when they show that the ``map method'' is capable of shadowing
the Lorenz equations for only 10 time units, while their method,
which rescales time, can find shadows lasting time units --
ten *thousand* times longer!

Finally, note that the non-shadowable example given in the tutorial
(*x*''=0, click here ) *is* obviously shadowable
if time is rescaled. This matches what our intuition would say: as
long as we only care about qualitative properties of the solution,
why should it matter that the numerical solution traverses the
path at a slightly different velocity than the true solution, as
long as the trajectories, as a whole, remain near to each other?

Fri Dec 27 17:41:39 EST 1996

Access count (updated once a day) since 1 Jan 1997: 8876