Systematics of the Stable Periods

Abstract

If we look closely at the Figure 1.19, we distinguish, from left to right, more or less clearly, a stable orbit of period 1 (i.e., a fixed point), then a stable orbit of period 2, then 4, 8, then a “mess,” then another orbit of period 6, then 5, and 3. Other stable periods are barely visible inbetween. The astonishing fact about this arrangement of stable periodic orbits is its independence of the particular one-parameter family of maps. A complete description of this arrangement will be given in III.1, based on earlier/developments in II.2. We describe here some of its aspects. Note first of all, that there are different kinds of stable periods of the same length. This differentiation comes from the order in which the points are visited. As an example, there are two different stable periods of length four (the second one is barely visible to the right of the period 3 in Fig. I.19).