On the Global Asymptotic Stability and 4-Period Oscillation of the Third-Order Difference Equation

IF 1.2 Q2 MATHEMATICS, APPLIED Journal of Applied Mathematics Pub Date : 2023-11-11 DOI:10.1155/2023/5726617

M. E. Erdogan

{"title":"On the Global Asymptotic Stability and 4-Period Oscillation of the Third-Order Difference Equation","authors":"M. E. Erdogan","doi":"10.1155/2023/5726617","DOIUrl":null,"url":null,"abstract":"The main objective of this paper is to study the global behavior and oscillation of the following third-order rational difference equation <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\"> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>α</mi> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>/</mo> <mi>β</mi> <msubsup> <mrow> <mi>x</mi> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>γ</mi> <msubsup> <mrow> <mi>x</mi> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </math> , where the initial conditions <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\"> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </math> are nonzero real numbers and <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\"> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> </math> are positive constants such that <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\"> <mi>α</mi> <mo>≤</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> </math> . Visual examples supporting solutions are given at the end of the study. The figures are found with the help of MATLAB.","PeriodicalId":49251,"journal":{"name":"Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Applied Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1155/2023/5726617","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

The main objective of this paper is to study the global behavior and oscillation of the following third-order rational difference equation

x_{n + 1} = α x_{n} x_{n - 1} x_{n - 2} / β x_{n - 1}^{2} + γ x_{n - 2}^{2}

, where the initial conditions

x_{- 2}, x_{- 1}, x_{0}

are nonzero real numbers and

α, β, γ

are positive constants such that

α \leq β + γ

. Visual examples supporting solutions are given at the end of the study. The figures are found with the help of MATLAB.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

三阶差分方程的全局渐近稳定性和4周期振荡

本文的主要目的是研究全球行为和振荡的三阶有理差分方程x n + 1 =αx n n−1 x n−2 /βx n−1 2 +γx n−2 2,初始条件x−2,x−1,x非零实数和α,β,γ是积极的常量,α≤β+γ。在研究的最后给出了支持解决方案的可视化示例。图形是借助MATLAB软件进行绘制的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Journal of Applied Mathematics MATHEMATICS, APPLIED-

CiteScore

2.70

自引率

0.00%

发文量

审稿时长

3.2 months

期刊介绍： Journal of Applied Mathematics is a refereed journal devoted to the publication of original research papers and review articles in all areas of applied, computational, and industrial mathematics.