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 . 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?