dynamics of a system of higher order difference equations with a period-two coefficient

The aim of this paper is to study the dynamics of the system of two rational difference equations: xn+1 = αn + yn−k/yn, yn+1 = αn + xn−k/xn, n = 0, 1, . . . where {αn}n≥0 is a two periodic sequence of nonnegative real numbers and the initial conditions xi , yi are arbitrary positive numbers for i = −k, −k + 1, −k + 2, . . . , 0 and k ∈ n. we investigate the boundedness character of positive solutions. in addition, we establish some sufficient conditions under which the local asymptotic stability and the global asymptotic stability are assured. furthermore, we determine the rate of the convergence of the solutions. some numerical are considered in order to confirm our theoretical results.