Local dynamics and global attractivity of a certain second-order quadratic fractional difference equation

Kulenović, Mustafa RS; Pilav, Esmir; Silić, Enesa

doi:10.1186/1687-1847-2014-68

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 5 Equations of type $(1, 3)$

From: Local dynamics and global attractivity of a certain second-order quadratic fractional difference equation

Equation	Equilibrium point	Stability of equilibrium point	Period-two solution and stability	Partial derivatives
$x_{n + 1} = \frac{x_{n - 1} x_{n}}{B x_{n} x_{n - 1} + C x_{n - 1}^{2} + D x_{n}}$	$\bar{x} = \frac{1 - D}{B + C}$ exists for D<1	LAS for D<1	no minimal period-two sol.	$f_{u}^{'} = \frac{C v^{3}}{{(v (B u + C v) + D u)}^{2}}$ $f_{v}^{'} = \frac{D u^{2} - C u v^{2}}{{(B u v + C v^{2} + D u)}^{2}}$
$x_{n + 1} = \frac{x_{n - 1}^{2}}{B x_{n} x_{n - 1} + C x_{n - 1}^{2} + D x_{n}}$	$\bar{x} = \frac{1 - D}{B + C}$ exists for D<1	LAS for $D < \frac{C - B}{B + 3 C}$ a saddle for $D > \frac{C - B}{B + 3 C}$ a non-hyp. for $D = \frac{C - B}{B + 3 C}$	${ϕ_{1}, ψ_{1}} = {0, 1 / C}$ -LAS ${ϕ_{2}, ψ_{2}}$ exists for $D < \frac{C - B}{B + 3 C}$ $ϕ_{2} = \frac{D + 1 + \frac{\sqrt{Δ}}{B - C}}{2 C}$ $ϕ_{2} = \frac{D + 1 - \frac{\sqrt{Δ}}{B - C}}{2 C}$ Δ = (D + 1)(B − C)(BD + B + 3CD − C) a saddle point for $D < \frac{C - B}{B + 3 C}$	$f_{u}^{'} = \frac{- v^{2} (B v + D)}{{(v (B u + C v) + D u)}^{2}}$ $f_{v}^{'} = \frac{B u v^{2} + 2 D u v}{{(B u v + C v^{2} + D u)}^{2}}$
$x_{n + 1} = \frac{x_{n}}{B x_{n} x_{n - 1} + C x_{n - 1}^{2} + D x_{n}}$	$\bar{x} = \frac{\sqrt{4 B + 4 C + D^{2}} - D}{2 (B + C)}$	LAS for $B > \frac{C^{2} - 2 C D^{2}}{D^{2}}$ a repeller for $B < \frac{C^{2} - 2 C D^{2}}{D^{2}}$ a non-hyp. eq. for $B = \frac{C^{2} - 2 C D^{2}}{D^{2}}$	no minimal period-two sol.	$f_{u}^{'} = \frac{C v^{2}}{{(v (B u + C v) + D u)}^{2}}$ $f_{v}^{'} = \frac{- u (B u + 2 C v)}{{(B u v + C v^{2} + D u)}^{2}}$

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