Dynamics of Two Extensive Classes of Recursive Sequences
Received:August 23, 2008  Revised:January 05, 2009
Key Word: recursive sequence   equilibrium   dynamics.
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.10771169).
 Author Name Affiliation Bin ZHAO Department of Mathematics, Northwest Agriculture & Forestry University, Shaanxi 712100, P. R. China Department of Mathematics, Northwest University, Shaanxi 710127, P. R. China
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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.