A pseudo-stable structure in a completely invertible bouncer system

Abstract

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.

Finite-Time Stabilization of Unstable Orbits in the Fractional Difference Logistic Map

“…H-ranks provide a straightforward but effective way to measure the algebraic complexity of a sequence. This measure has been successfully employed to analyze complexity in a number of different maps such as the invertible logistic map [30], the bouncer map [31], the Gauss map [32], as well as the fractional logistic map [33].…”

The Order of a 2-Sequence and the Complexity of Digital Images

“…Thus, for real-world applications it is recommended to choose ε > 0 and investigate the pseudo-order. A number of algorithms that use the concept of 1-LRS have successfully applied this approach [15][16][17]. It is clear that an alternative approach is also required for the computation of 2-LRS orders.…”

“…The shorter average period results into a smaller LRS pseudo-order. 2-LRS pseudo-order for the image of bricks (without noise) is equal to (22,17) corresponding to ( 1 22 , 1 17 ) in Fig. 2 at λ = 1.…”