About Stable Periodic Helixes, L-iteration and Chaos Generated by Unbounded Functions

Abstract

We consider stable periodic helixes as a generalization of stable periodic orbits. We see that in the studied class of iterated functions Chaos always arise suddenly. Therefore, we shall study the route from chaos to order rather than the route from order to chaos. We show that, paradoxically, genuine Chaos may look as much like Order and during as many iteration steps as one may wish. Then, we shall propose a generalization of the idea of map iteration that do not imply the existence of periodic orbits. We shall show that, within a strictly deterministic context, unpredictability, aperiodic order, sensitive dependence and chaos are completely different concepts and we shall try to show what this difference is made of. We shall also propose an example of non chaotic aperiodic order.