Second Minimal Orbits, Sharkovski Ordering and Universality in Chaos

This paper introduces the notion of second minimal [Formula: see text]-periodic orbits of continuous maps on the interval according to whether [Formula: see text] is a successor of the minimal period of the map in the Sharkovski ordering. We pursue the classification of second minimal [Formula: see text]-orbits in terms of cyclic permutations and digraphs. It is proven that there are nine types of second minimal 7-orbits with accuracy up to inverses. The result is applied to the problem of the distribution of periodic windows within the chaotic regime of the bifurcation diagram of the one-parameter family of unimodal maps. It is revealed that by fixing the maximum number of appearances of periodic windows, there is a universal pattern of distribution. In particular, the first appearance of all the orbits is always a minimal orbit, while the second appearance is a second minimal orbit. It is observed that the second appearance of the 7-orbit is a second minimal 7-orbit with a Type 1 digraph. The reason for the relevance of the Type 1 second minimal orbit is the fact that the topological structure of the unimodal map with a single maximum, is equivalent to the structure of the Type 1 piecewise monotonic endomorphism associated with the second minimal 7-orbit. Yet another important report of this paper is the revelation of universal pattern dynamics with respect to an increased number of appearances.