Floating-point shadowing for 2D saddle-connection

CDSNS Colloquium
Monday, October 28, 2013 - 15:05
1 hour (actually 50 minutes)
Skiles 202
Chebyshev laboratory, Saint-Petersburg
There is known a lot of information about classical or standard shadowing. Itis also often called a pseudo-orbit tracing property (POTP).  Let M be a closedRiemannian manifold. Diffeomorphism f : M → M is said to have POTPif for a given accuracy any pseudotrajectory with errors small enough can beapproximated (shadowed) by an exact trajectory. Therefore, if one wants to dosome numerical investiagion of the system one would definitely prefer it to haveshadowing property.However, now it is widely accepted that good (qualitatively strong) shad-owing is present only in hyperbolic situations. However it seems that manynonhyperbolic 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. Onecan consider only pseudotrajectories that resemble sequences of points generatedby a computer with floating-point arithmetic.I will tell what happens in the (simplified) case of “linear” two-dimensionalsaddle connection. In this case even stochastic versions of classical shadowing(when one tries to ask only for most pseudotrajectories to be shadowed) do notwork. Nevertheless, for “floating-point” pseudotrajectories one can prove somepositive results.There is a dichotomy: either every pseudotrajectory stays close to the un-perturbed trajectory forever if one carefully chooses the dependence betweenthe size of errors and requested accuracy of shadowing, or there is always apseudotrajectory that can not be shadowed.