Dynamics and behaviors of a third-order system of difference equation

Motivated by an Conjecture in the literature “Dynamics of Second Order Rational Difference Equations with Open Problems and Conjecture”, we introduce a difference equation system: xn+1 = yn + yn−2 xn−1 , yn+1 = xn + xn−2 yn−1 where xi, yi ∈ (0,∞), i ∈ {−2,−1, 0}. If the initial value satisfy xi = yi, i = {−2,−1, 0}, then the system reduces into the Conjecture. In this paper, we investigate the asymptotic behaviors, periodicity and oscillatory of the system.