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A new bifurcation case for the cycle length is found in this paper for rational difference equations, which is shown out from the following fifth order rational difference equation a x x x x x x a x x x x x x x n n n n n n n n n n n n n + + + 0  x . Mainly, the perturbation of the initial values may lead to the essential variation of the cycle length rule for the nontrivial solutions of the equation. That is, with the change of the initial values, the successive lengths of positive and negative semicycles for nontrivial solutions of this equation is found to periodically occur with multiple different prime periods, respectively, 4 ,12 . Furthermore, in any one fixed period, the successive occurring order of positive and negative semicycles is completely inverse, i.e., for the period 4 ,t he order is either + 3 ,  1 or  3 , + 1 in a period, and for the period 12 , the order is either +

The Bifurcation of Cycle Length and Global Asymptotic Stability in a Rational Difference Equation with Higher Order