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

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Abstract

In recent years, there is a wide interest in Sarkovskii's theorem and 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. But f can not have all n-periodic orbits for some n.

Example. Let

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

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

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References

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Authors and Affiliations

  1. Chengdu Branch, Academia Sinica, Chengdu

    Zhang Jing-zhong, Yang Lu & Zhang Lei

Authors
  1. Zhang Jing-zhong
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  2. Yang Lu
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  3. Zhang Lei
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Additional information

Communicated by Chien Wei-zang

Project Supported by the National Natural Science Foundation of Science of China.

The criterion algorithm of relation of implication between periodic orbits (I), Applied Mathematics and Mechanics, 10, 11 (1989), 1029–1037.

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Jing-zhong, Z., Lu, Y. & Lei, Z. The criterion algorithm of relation of implication between periodic orbits (II). Appl Math Mech 11, 139–147 (1990). https://doi.org/10.1007/BF02014538

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  • DOI: https://doi.org/10.1007/BF02014538

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