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Table 1 Equations of type (1,2)

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 = β x n 1 x n C x n 1 2 + x n x ¯ = β 1 C for β>1
no eq. point for β ≤ 1
LAS for β>1 no period-two solution f u = C v 3 β ( C v 2 + u ) 2
f v = u β ( u C v 2 ) ( C v 2 + u ) 2
x n + 1 = γ x n 1 2 C x n 1 2 + x n x ¯ = γ 1 C for γ>1
no eq. point for γ ≤ 1
LAS for γ>3
a saddle point for 1<γ<3
a non-hyp. eq. for γ = 3
{0,γ/C}-LAS
{ γ + 1 ( γ 3 ) ( γ + 1 ) 2 C , γ + 1 + ( γ 3 ) ( γ + 1 ) 2 C }
a saddle point for γ>3
a non-hyp. eq. for γ = 3
f u = γ v 2 ( C v 2 + u ) 2
f v = 2 γ u v ( C v 2 + u ) 2
x n + 1 = γ x n 1 2 B x n 1 x n + x n x ¯ = γ 1 B for γ>1
no eq. point for γ ≤ 1
a saddle point for γ>1 no period-two solution f u = γ v 2 u 2 ( B v + 1 )
f v = γ v ( B v + 2 ) u ( B v + 1 ) 2
x n + 1 = δ x n B x n x n 1 + x n 1 2 x ¯ = δ B + 1 a repeller for δ>0, B>0 no period-two solution f u = δ ( B u + v ) 2
f v = u δ ( B u + 2 v ) ( B u v + v 2 ) 2
x n + 1 = δ x n C x n 1 2 + x n x ¯ = 4 C δ + 1 1 2 C LAS for <2
a repeller for >2
a non-hyp. eq. for  = 2
no period-two solution f u = C v 2 δ ( C v 2 + u ) 2
f v = 2 C u v δ ( C v 2 + u ) 2