Dynamics of Two Extensive Classes of Recursive Sequences

DOI：10.3770/j.issn:1000-341X.2010.05.022

 作者 单位 赵斌 西北大学数学系, 陕西 西安 710127 西北农林科技大学数学系, 陕西 杨凌 712100

研究了两类广泛递归序列:$$x_{n 1}={c\sum\limits_{j=0}^{k}\sum\limits_{( i_0,i_1,\ldots, i_{2j})\in A_{2j}}x_{n-i_0}x_{n-i_1}\cdots x_{n-i_{2j}} f(x_{n-i_0}, x_{n-i_1}, \ldots, x_{n-i_{2k}})\over c\sum\limits_{j=1}^{k}\sum\limits_{( i_0,i_1,\ldots, i_{2j-1})\in A_{2j-1}}x_{n-i_0}x_{n-i_1}\cdots x_{n-i_{2j-1}} c f(x_{n-i_0}, x_{n-i_1}, \ldots, x_{n-i_{2k}})}$$和$$x_{n 1}={c\sum\limits_{j=1}^{k}\sum\limits_{( i_0, i_1,\ldots, i_{2j-1})\inA_{2j-1}}x_{n-i_0}x_{n-i_1}\cdots x_{n-i_{2j-1}} c f(x_{n-i_0}, x_{n-i_1}, \ldots, x_{n-i_{2k}})\over c\sum\limits_{j=0}^{k}\sum\limits_{( i_0, i_1,\ldots, i_{2j})\in A_{2j}}x_{n-i_0}x_{n-i_1}\cdots x_{n-i_{2j}} f(x_{n-i_0}, x_{n-i_1}, \ldots, x_{n-i_{2k}})}$$的动力特性, 证明了它们的唯一正平衡点$\overline{x} =1$是全局渐近稳定的, 为递归序列理论的研究提供了一种新的途径.

We investigate the dynamics of two extensive classes of recursive sequences:$$x_{n 1}={c\sum\limits_{j=0}^{k}\sum\limits_{( i_0,i_1,\ldots, i_{2j})\in A_{2j}}x_{n-i_0}x_{n-i_1}\cdots x_{n-i_{2j}} f(x_{n-i_0}, x_{n-i_1}, \ldots, x_{n-i_{2k}})\over c\sum\limits_{j=1}^{k}\sum\limits_{( i_0,i_1,\ldots, i_{2j-1})\in A_{2j-1}}x_{n-i_0}x_{n-i_1}\cdots x_{n-i_{2j-1}} c f(x_{n-i_0}, x_{n-i_1}, \ldots, x_{n-i_{2k}})},$$ and $$x_{n 1}={c\sum\limits_{j=1}^{k}\sum\limits_{( i_0, i_1,\ldots, i_{2j-1})\in A_{2j-1}}x_{n-i_0}x_{n-i_1}\cdots x_{n-i_{2j-1}} c f(x_{n-i_0}, x_{n-i_1}, \ldots, x_{n-i_{2k}})\over c\sum\limits_{j=0}^{k}\sum\limits_{( i_0, i_1,\ldots, i_{2j})\in A_{2j}}x_{n-i_0}x_{n-i_1}\cdots x_{n-i_{2j}} f(x_{n-i_0}, x_{n-i_1}, \ldots, x_{n-i_{2k}})}.$$ We prove that their unique positive equilibrium $\overline{x} =1$ is globally asymptotically stable. And a new access is presented to study the theory of recursive sequences.