﻿ 差分方程xn+1=xn/(p+xn-1)的动力学性质 Dynamics of the Difference Equation xn+1=xn/(p+xn-1)

Vol. 07  No. 11 ( 2018 ), Article ID: 27627 , 3 pages
10.12677/AAM.2018.711163

Dynamics of the Difference Equation ${x}_{n+1}={x}_{n}/\left(p+{x}_{n-1}\right)$

Shaogao Deng1, Lijun Zhu2*

1School of Mathematics, Southwest Jiaotong University, Chengdu Sichuan

2School of Mathematics and Information Science, North Minzu University, Yinchuan Ningxia

Received: Oct. 23rd, 2018; accepted: Nov. 13th, 2018; published: Nov. 20th, 2018

ABSTRACT

This paper considers the difference equation ${x}_{n+1}={x}_{n}/\left(p+{x}_{n-1}\right)\left(p\ge 0,n\ge 2\right)$ with the initial values ${x}_{1}>0,{x}_{2}>0$ . The asymptotic stability of the positive solutions is proved under some assumptions.

Keywords:Difference Equation, Equilibrium Point, Asymptotic Stability

1西南交通大学数学学院，四川 成都

2北方民族大学数学与信息科学学院，宁夏 银川

1. 引言

${x}_{n+1}={x}_{n}/\left(p+{x}_{n-1}\right)$ (1.1)

${x}_{n+1}=f\left({x}_{n},{x}_{n-1}\right)\left(n\ge 2\right)$ (1.2)

${\lambda }^{2}-{f}_{x}\left(\beta ,\beta \right)\lambda -{f}_{y}\left(\beta ,\beta \right)=0$ (1.3)

2. 主要结果

1) 当 $p\ge 1$ 时，有唯一的平衡点 $\beta =0$ ，且是全局(渐近)稳定的；

2) 当 $0 时，有两个平衡点 $\beta =0$$\beta =1-p$ ，其中 $\beta =0$ 是不稳定的，而 $\beta =1-p$ 是(渐近)稳定的。

1) 当 $p\ge 1,n\ge 2$ 时， ${x}_{n+1}/{x}_{n}=1/\left(p+{x}_{n-1}\right)<1/p\le 1$

$\beta =0或\beta =1-p\left(当p>1时舍去\right)$

2) 当 $0 时，令 ${x}_{n}\equiv \beta$ ，可得：

$\beta =0或\beta =1-p>0$

$f\left(x,y\right)=x/\left(p+y\right)$ ，则：

${f}_{x}\left(x,y\right)=1/\left(p+y\right)$${f}_{y}\left(x,y\right)={-x/\left(p+y\right)}^{2}$

$\beta =0$ 时， ${f}_{x}\left(\beta ,\beta \right)=1/p$${f}_{y}\left(\beta ,\beta \right)=0$

${\lambda }_{1}=0,{\lambda }_{2}=1/p>1$

$\beta =1-p$ 时， ${f}_{x}\left(\beta ,\beta \right)=1$${f}_{y}\left(\beta ,\beta \right)=p-1$

$0<{\lambda }_{1}=\left(1-\sqrt{4p-3}\right)/2\le {\lambda }_{2}=\left(1+\sqrt{4p-3}\right)/2<1\left(当3/4\le p<1时\right)$

${‖{\lambda }_{1}‖}^{2}={‖{\lambda }_{2}‖}^{2}={\lambda }_{1}{\lambda }_{2}=1-p<1\left(当0

${x}_{3}=b/a,{x}_{4}=1/a,{x}_{5}=1/b,{x}_{6}=a/b,{x}_{7}=a,{x}_{8}=b\cdots \cdots$

Dynamics of the Difference Equation xn+1=xn/(p+xn-1)[J]. 应用数学进展, 2018, 07(11): 1402-1404. https://doi.org/10.12677/AAM.2018.711163

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6. NOTES

*通讯作者。