Bifurcation and periodically semicycles for fractional difference equation of fifth order

Our paper takes into account a new bifurcation case of the cycle length and a fifth-order difference equation dynamics of { Mathematical Formulas } where γ ∈ (0, ∞) , α, β ∈ Z+, and y−4, y−3, y−1, y−2, y0 ∈ (0, ∞) is took into consideration. The disturbance of initials lead to a distinction of cycle length principle of the non-trivial solutions of the equation. The principle of the track solutions structure for this equation is given. The consecutive periods of negative and positive semicycles of non-trivial solutions of this equation take place periodically with only prime period fifteen and in a period with the principles represented by either {3+, 1−, 2+, 2−, 1+, 1−, 1+, 4−} or {3−, 1+, 2−, 2+, 1−, 1+, 1−, 4+}. From this rubric we will establish that the positive fixed point has global asymptotic stability.