Composition law of cardinal ordering permutations

Abstract

a b s t r a c tIn this paper the theorems that determine composition laws both cardinal ordering permutations and their inverses are proven. So, the relative position of points in an hs-periodic orbit is completely known as well as which order those points are visited, no matter how the hs-periodic orbit emerges, be it through a period doubling cascade (s = 2 n ) of as the h-periodic orbit or a primary window (like saddle-node bifurcation cascade with h = 2 n ) or a secondary window (the birth of an s-periodic window inside the h-periodic one). Certainly, period doubling cascade orbits are particular cases with h = 2 and s = 2 n . Both composition laws are also shown in algorithmic way for easy use.

Compound orbits break-up in constituents: An algorithm

“…If we had a compound hs-periodic orbit we could decompose it in two orbits of periods h and s, respectively, according to theorem 1. This process is the opposite to that described in [9] where two orbits with periods h and s were composed to generate an hs-periodic orbit. Therefore, theorem 1 closes the theoretical frame of composition and decomposition.…”

“…An s-periodic orbit inside the h-periodic window, must follow a visiting order in its Poincaré map that can be decomposed using decomposition Theorem 1: from a known periodic orbit, another two unique orbits can be described. This link between periodic orbits (not only from simpler to more complex, as studied in [9], but also from complex to simpler orbits, as studied in this paper) imposes strong restrictions to a physical system dependent on one control parameter, whose underlying origin must be studied. Notice that the cardinality of the sets A and B are respectively (n − 1)s + r and (h − n)s − (r − 1).…”

“…. , I h−1 is even (definition 2 [9]), so f h has a maximum (lemma 3 [9]) and then the link of a point of the central box with the next visiting point in this same box is given by β s , as we have just explained above. But the linking of these two points requires visiting all boxes before they connect between themselves.…”

“…The geometric meaning of composing O h and O s involves replacing the h points of O h by h boxes, with s points each, such that all points of a same box are mapped into the same box. It is important to point out that boxes are visited consecutively according to β h and that every time the same box is visited then the box points are visited according to β s , if f h has a maximum and according to β s if f h has a minimum (see [9] for more details). As boxes (see definition 3) H i with i = 1, .…”

“…c.1) c is odd. As C is odd the number of points of O h located in J R is even (in [9] this is said as the R-parity of I 1 , . .…”