The dynamics and behavior of logarithmic type fuzzy difference equation of order two

by Muhammad Usman, Abdul Khaliq, Muhammad Azeem, Senesie Swaray, Mohamed Kallel

Fuzzy difference equations are becoming increasingly popular in fields like engineering, ecology, and social science. Difference equations find numerous applications in real-life problems. Our study demonstrates that the logarithmic-type fuzzy difference equation of order two possesses a nonnegative solution, and an equilibrium point, and exhibits asymptotic behavior. x n + 1 = α + β l o g x n A + x n - 1 , ∀ n ∈ W Where, x_i represents the sequence of fuzzy numbers, and the parameters α, β, A, along with the initial conditions x₋₁ and x₀, are positive fuzzy numbers. The characterization theorem is employed to convert each single logarithmic fuzzy difference equation into a set of two crisp logarithmic difference equations within a fuzzy environment. We evaluated the stability of the equilibrium point of the fuzzy system. Utilizing variational iteration techniques, the method of g-division, inequality skills, and a theory of comparison for logarithmic fuzzy difference equations, we investigated the governing equation dynamics, including its boundedness, existence, and both local and global stability analysis. Additionally, we provided some numerical solutions for the equation describing the system to verify our results.