There is known a lot of information about classical or standard shadowing. It is also often called a pseudo-orbit tracing property (POTP). Let M be a closed Riemannian manifold. Diffeomorphism $f : M \to M$ is said to have POTP if for a given accuracy any pseudotrajectory with errors small enough can be approximated (shadowed) by an exact trajectory. Therefore, if one wants to do some numerical investiagion of the system he would definitely prefer it to have shadowing property.

However, now it is widely accepted that good (qualitatively strong) shad- owing is present only in hyperbolic situations. However it seems that many nonhyperbolic systems still could be well analysed numerically.

As a step to resolve this contradiction I introduce some sort of weaker shad- owing. The idea is to restrict a set of pseudotrajectories to be shadowed. One can consider only pseudotrajectories that resemble sequences of points generated by a computer with floating-point arithmetic.

I will tell what happens in the (simplified) case of “linear” two-dimensional saddle connection. In this case even stochastic versions of classical shadowing (when one tries to ask only for most pseudotrajectories to be shadowed) do not work. Nevertheless, for “floating-point” pseudotrajectories one can prove some positive results.

There is a dichotomy: either every pseudotrajectory stays close to the un-perturbed trajectory forever if one carefully chooses the dependence between the size of errors and requested accuracy of shadowing, or there is always a pseudotrajectory that can not be shadowed.