The criterion algorithm of relation of implication between periodic orbits (I)

Abstract

In recent years, there is a wide interest in Sarkovskii's theorem and the related study. According to Sarkovskii's theorem if the continuous self-mapf of the closed interval has a 3-periodic orbit, thenf must has an n-periodic orbit for any positive integer n. Butf can not has all n-periodic orbits for some n. For example, let

$$f(x) = \left\{ {\begin{array}{*{20}c} {x + \frac{1}{2}} & {\left( {x \in \left[ {0, \frac{1}{2}} \right]} \right)} \\ {2(1 - x)} & {\left( {x \in \left[ {\frac{1}{2}, 1} \right]} \right)} \\ \end{array} } \right.$$

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Evidently,f has only one kind of 3-periodic orbit in the two kinds of 3-periodic orbits. This explains that it isn't far enough to uncover the relation between periodic orbits by information which Sarkovskii's theorem has offered. In this paper, we raise the concept of type of p periodic orbits, and give a feasible algorithm which decides the relation of implication between two periodic orbits.