On the lower bound error for discrete maps using associative property

This paper introduces a class of pseudo-orbits which guarantees the same lower bound error (LBE) for two different natural interval extensions of discrete maps. In previous work, the LBE was investigated along with a simple technique to evaluate numerical accuracy of free-run simulations of polynomial NARMAX or similar discrete maps. Here we prove that it is possible to calculate the LBE for two pseudo-orbits, extending so the results of previous work in which the LBE is valid for only one of the two pseudo-orbits. The main application of this technique is to provide a simple estimation of the LBE. We illustrate our approach with the Logistic Map and Hénon Map. Using double precision, our results show that we ought simulate the Logistic Map and Hénon Map with less than 100 iterations, which is, for instance, far less than the number usually considered as transient to build bifurcation diagrams.