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Our paper takes into account a new bifurcation case of the cycle length and a fifth-order difference equation dynamics of ym+1 = ymy α m−2y β m−4 + ym + y α m−2 + y β m−4 + γ ymy α m−2 + y α m−2y β m−4 + ymy β m−4 + γ+ 1 , m = 0, 1, 2, 3, . . . , 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.

Bifurcation and periodically semicycles for fractional difference equation of fifth order