<p>In this article, we examine the existence, uniqueness and qualitative characteristics of the solutions to the following two-dimensional symmetric exponential-type non-linear fuzzy difference equations (FDE) system with second-order <Equation ID="Equ128"> <EquationSource Format="TEX">$\begin{aligned} {\left\{ \begin{array}{ll} x_{n+1}& =\alpha _{1}+\beta _{1}x_{n-1}+\gamma _{1}x_{n-1}e^{-y_{n}}\\ y_{n+1}& =\alpha _{2}+\beta _{2}y_{n-1}+\gamma _{2}y_{n-1}e^{-x_{n}} \end{array}\right. }, \quad n\in \mathbb {N}_{0}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <msub> <mi>x</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd columnalign="left"> <mrow> <mo>=</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <msub> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mn>1</mn> </msub> <msub> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>y</mi> <mi>n</mi> </msub> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <msub> <mi>y</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo>=</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <msub> <mi>y</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mn>2</mn> </msub> <msub> <mi>y</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> <mo>,</mo> <mspace width="1em" /> <mi>n</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">N</mi> <mn>0</mn> </msub> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where the positive fuzzy initial conditions are <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$x_{-i},y_{-i}, \ i \in \{0,1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mrow> <mo>-</mo> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow> <mo>-</mo> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="4pt" /> <mi>i</mi> <mo>∈</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha _{1}, \alpha _{2}, \beta _{1}, \beta _{2}, \gamma _{1}, \gamma _{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>γ</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> are positive fuzzy numbers. In addition to the theoretical components that make up the results, we provide several numerical examples.</p>

The dynamics of a second-order symmetric system of exponential fuzzy difference equations

期刊论文
发表日期：

2025年10月16日

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Sevda Atpinar,
Yasin Yazlik,
Qianhong Zhang

摘要

In this article, we examine the existence, uniqueness and qualitative characteristics of the solutions to the following two-dimensional symmetric exponential-type non-linear fuzzy difference equations (FDE) system with second-order $\begin{aligned} {\left\{ \begin{array}{ll} x_{n+1}& =\alpha _{1}+\beta _{1}x_{n-1}+\gamma _{1}x_{n-1}e^{-y_{n}}\\ y_{n+1}& =\alpha _{2}+\beta _{2}y_{n-1}+\gamma _{2}y_{n-1}e^{-x_{n}} \end{array}\right. }, \quad n\in \mathbb {N}_{0}, \end{aligned}$ $\begin{matrix} \{\begin{matrix} x_{n + 1} & = α_{1} + β_{1} x_{n - 1} + γ_{1} x_{n - 1} e^{- y_{n}} \\ y_{n + 1} & = α_{2} + β_{2} y_{n - 1} + γ_{2} y_{n - 1} e^{- x_{n}} \end{matrix}), n \in N_{0}, \end{matrix}$ where the positive fuzzy initial conditions are $x_{-i},y_{-i}, \ i \in \{0,1\}$ $x_{- i}, y_{- i}, i \in {0, 1}$ , and $\alpha _{1}, \alpha _{2}, \beta _{1}, \beta _{2}, \gamma _{1}, \gamma _{2}$ $α_{1}, α_{2}, β_{1}, β_{2}, γ_{1}, γ_{2}$ are positive fuzzy numbers. In addition to the theoretical components that make up the results, we provide several numerical examples.