DYNAMICS OF A HIGH-ORDER NONLINEAR FUZZY DIFFERENCE EQUATION

Abstract

This paper is concerned with the following high-order nonlinear fuzzy difference system $ {x_{n + 1}} = \frac{{A{\kern 1pt} {x_{n - m}}}}{{B + C{\kern 1pt} \prod\limits_{i = 0}^m {{x_{n - i}}} }}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n = 0, 1, 2, \cdots , $ where $ {x_n} $ is a sequence of positive fuzzy numbers, the parameters and the initial conditions $ {x_{ - m}}, \;{x_{ - m + 1}}, \; \cdots , {x_0} $ are positive fuzzy numbers, $ m $ is non-negative integer. More accurately, our main purpose is to study the existence and uniqueness of the positive solutions, the boundedness of the positive solutions, the instability, local asymptotic stability and global asymptotic stability of the equilibrium points for the above equation by using the iteration method, the inequality skills, the mathematical induction, and the monotone boundedness theorem. Moreover, some numerical examples to the difference system are given to verify our theoretical results.