A pseudo-stable structure in a completely invertible bouncer system

It is shown that a pseudo-stable structure of non-asymptotic convergence may exist in a completely invertible bouncing ball model. Visualization of the pattern of H-ranks helps to identify this structure. It appears that this structure is similar to the stable manifold of non-invertible nonlinear maps which govern the non-asymptotic convergence to unstable periodic orbits. But this convergence to the unstable repeller of the bouncing ball problem is only temporary since non-asymptotic convergence cannot exist in completely invertible maps. This nonlinear effect is exploited for temporary stabilization of unstable periodic orbits in completely reversible nonlinear maps.