一种计算Tchebyscheff映射高阶关联函数的数论方法  

A number theoretical method for calculating high order correlation functions of the Tchebyscheff maps

作　　者：周兴旺[1] ZHOU Xing-Wang(School of Mathematics,Sichuan University,Chengdu 610064,China)

机构地区：[1]四川大学数学学院,成都610064

出　　处：《四川大学学报（自然科学版）》2022年第5期33-38,共6页Journal of Sichuan University(Natural Science Edition)

基　　金：国家自然科学基金(12171335);桥梁无损检测与工程计算四川省高校重点实验室开放项目(2022QYJ07)。

摘　　要：关联函数是混沌映射的统计理论的核心.本文主要研究Tchebyscheff映射的高阶关联函数的计算问题.对此问题,已有Beck于1991年提出的一种图论方法.然而,当映射和关联函数的阶都比较大时该方法非常低效.本文基于Tchebyscheff映射关联函数的定义提出了一种数论方法.该方法将关联函数的计算问题转化为一类具有严格单调递增指数的指数型丢番图方程的求解问题,进而逐级地求得方程的解.然后,本文研究了当映射的阶不小于关联函数的阶时非零关联函数的计算问题.计算结果显示,此时关联函数的值不依赖于映射的阶,且非零关联函数的个数与第二类斯特林数密切相关.作为应用本文最后计算了满足条件的所有12阶非零关联函数的值.Correlation functions play a key role in the statistical description of chaotic maps. The main concern of this paper is the calculation of correlation functions of the Tchebyscheff maps, which is traditionally handled by using a graph theoretical method introduced by Beck in 1991. However, this method has poor efficiency when the orders of map and correlation function are large. To overcome this problem, we introduce a number theoretical method based on the definition of correlation functions of the Tchebyscheff maps. In this method, the calculation is transformed into solving a class of Diophantine equations with strictly increasing exponentials, which can be solved in a hierarchical way. Then we obtain all non-vanishing correlation functions with order not more than the order of map and show that the value of correlation functions is independent of the order of map as well as the number of non-vanishing correlation functions is closely related to the Stirling numbers of the second kind. As an application, we calculate all non-vanishing 12-order correlation functions of the maps with order no less than 12.

关 键 词：关联函数 Tchebyscheff映射 指数型丢番图方程 

分 类 号：O156.7[理学—数学]