Global Dynamics of Sixth-Order Fuzzy Difference Equation

Across many fields, such as engineering, ecology, and social science, fuzzy differences are becoming more widely used; there is a wide variety of applications for difference equations in real-life problems. Our study shows that the fuzzy difference equation of sixth order has a nonnegative solution, an equilibrium point and asymptotic behavior. <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfenced open="(" close=")" separators="|"> <mrow> <mrow> <mrow> <mi>D</mi> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mi>i</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mrow> <mo>/</mo> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi>E</mi> <mo>+</mo> <mi>F</mi> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mi>i</mi> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mi>G</mi> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mi>i</mi> <mo>−</mo> <mn>4</mn> </mrow> </msub> <mo>+</mo> <mi>H</mi> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mi>i</mi> <mo>−</mo> <mn>5</mn> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </mrow> </mfenced> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>0,1,2</mn> <mo>,</mo> <mo>…</mo> </math> , where <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </math> is the sequence of fuzzy numbers and the parameter <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mi>D</mi> <mo>,</mo> <mi>E</mi> <mo>,</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo>,</mo> <mi>H</mi> </math> and the initial condition <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </math> are nonnegative fuzzy number. The characterization theorem is used to convert each single fuzzy difference equation into a set of two crisp difference equations in a fuzzy environment. So, a pair of crisp difference equations is formed by converting the difference equation. The stability of the equilibrium point of a fuzzy system has been evaluated. By using variational iteration techniques and inequality skills as well as a theory of comparison for fuzzy difference equations, we investigated the governing equation dynamics, such as its boundedness, existence, and local and global stability analysis. In addition, we provide some numerical solutions for the equation describing the system to verify our results.

Topic: Mathematical and Theoretical Epidemiology and Ecology Models
Subfield: Public Health, Environmental and Occupational Health
Field: Medicine
Domain: Health Sciences