切比雪夫映射族关联函数的指数型丢番图方程

摘要：切比雪夫映射族是一类典型的混沌映射，关联函数是研究其统计性质的关键. 本文所研究的指数型丢番图方程源于该映射族的关联函数的计算问题. 为求得该方程的解，本文首先对该方程进行简化，使简化后的方程具有严格单调递增的指数及非零系数. 然后本文引入了“块”的概念，根据简化方程所含块的个数对其进行了分类，进而将原丢番图方程求解问题转化为由块所构成的丢番图方程的求解问题. 本文最后研究了一个和两个块的情形，并举例说明了本文结果的应用.

关键词：指数型丢番图方程； 关联函数； 切比雪夫映射

收稿日期： 2022-11-18

基金项目： 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））

作者简介： 周兴旺（1973-），四川邛崃人，博士，主要研究方向为动力系统理论与应用.E-mail： zhouxw@scu.edu.cn

Exponential Diophantine equations for

correlation functions of Tchebyscheff maps

ZHOU Xing-Wang

（School of Mathematics， Sichuan University， Chengdu 610064， China）

Tchebyscheff map is a typical chaotic map and the correlation functions play a key role in the study of its statistical properties. This paper aims at the solutions of a class of exponential Diophantine equations arising in the calculation of correlation functions of the Tchebyscheff map. To solve the equation， we firstly reduce it to a new Diophantine equation with strictly increasing exponentials and nonzero coefficients. Then we introduce the definition of "block" and classify the reduced equations based on the number of blocks constructing the equation， thus transform the problem of solving the  equation to solve the new equations constructed by blocks. Finally we solve the equations constructed by one block and two blocks and exemplify the application of the main results.

Exponential Diophantine equation; Correlation function; Tchebyscheff map

（2010 MSC 15L06）

1 Introduction

Chaotic maps appear in many practical models， such as dynamical system^[1-4]， optimization theory^[5]， signal processing^[6]， communication^[7]， medicine^[8]， etc. As the key of statistical properties of chaotic maps^[9]， the calculation of （non-vanishing）correlation functions is an important but difficult problem in general， especially for the high-order correlation functions.

Our main concern in this paper is the calculation of correlation functions of the Tchebyscheff maps^[10-12]. Please recall that the correlation function of order m≥1 of an arbitrary Tchebyscheff map T_N≥2is defined as

6 Conclusions

This paper has considered the solutions of an exponential Diophantine equation resulting from the calculation of correlation functions of Tchebyscheff map T_N≥2. After reducing the original equation to a new Diophantine equation with strictly increasing exponentials and non-zero coefficients， we introduce the definition of block and classify the reduced equations based on the number of blocks constructing the equation. Then we start to solve the reduced equations constructed by one and two blocks. An example is proposed as the application of the main results. At the end of this paper， we should point that the solutions of the exponential Diophantine equation becomes more and more complex with the increase of order due to the effect of "combinations explosion".

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引用本文格式：

中 文： 周兴旺. 切比雪夫映射族关联函数的指数型丢番图方程[J]. 四川大学学报： 自然科学版， 2023， 60： 061006.

英 文： Zhou X W. Exponential Diophantine equations for correlation functions of Tchebyscheff maps [J]. J Sichuan Univ： Nat Sci Ed， 2023， 60： 061006.