Lately the study of chaotic dynamical systems with a computer has become a very active and popular field. An important question is, whether the orbits that were calculated and hence perturbed by roundoff errors and the exact orbits correspond approximately. There are some answers for the case of hyperbolic Axiom A functions. This paper gives a positive answer for the case of the famous quadratic family \(f_{\mu}(x)=\mu x(1-x)\) and the tent family \(f_{\mu}(x)=IF\) \(x\leq 1/2\) THEN \(\mu\) x ELSE \(\mu\) (1- x).
The authors show, that for points of certain sets the corresponding numerical orbits approximate the exact orbit, provided the numerical perturbation is small enough and the parameter \(\mu\) is slightly increased - with the exception of a countable number of parameter values. A counterexample illustrates the breakdown of that property.
The proof uses kneading theory by Milnor and Thurston and mainly relies on the fact that the itineraries of the considered maps are stable under perturbation - at least with an increased parameter value. The basics and the necessary theorems of symbolic dynamics are outlined nicely in the paper. A generalization from the quadratic resp. the tent maps to their respective topological and kneading properties and a generalization for maps with several critical points concludes the paper.