Dynamical Properties for a Class of Fourth-Order Nonlinear Difference Equations

We consider the dynamical properties for a kind of fourth-order rational difference equations. The key is for us to find that the successive lengths of positive and negative semicycles for nontrivial solutions of this equation periodically occur with same prime period 5. Although the period is same, the order for the successive lengths of positive and negative semicycles is completely different. The rule is . . . , 3 , 2−, 3 , 2−, 3 , 2−, 3 , 2−, . . . , or . . . , 2 , 1−, 1 , 1−, 2 , 1−, 1 , 1−, . . . , or . . . , 1 , 4−, 1 , 4−, 1 , 4−, 1 , 4−, . . . . By the use of the rule, the positive equilibrium point of this equation is proved to be globally asymptotically stable.