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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 = x n 1 x n B x n x n 1 + C x n 1 2 + D x n x ¯ = 1 D B + C exists for D<1 LAS for D<1 no minimal period-two sol. f u = C v 3 ( v ( B u + C v ) + D u ) 2
f v = D u 2 C u v 2 ( B u v + C v 2 + D u ) 2
x n + 1 = x n 1 2 B x n x n 1 + C x n 1 2 + D x n x ¯ = 1 D B + C exists for D<1 LAS for D< C B B + 3 C
a saddle for D> C B B + 3 C
a non-hyp. for D= C B B + 3 C
{ ϕ 1 , ψ 1 }={0,1/C}-LAS
{ ϕ 2 , ψ 2 } exists for D< C B B + 3 C
ϕ 2 = D + 1 + Δ B C 2 C
ϕ 2 = D + 1 Δ B C 2 C
Δ = (D + 1)(B − C)(BD + B + 3CD − C)
a saddle point for D< C B B + 3 C
f u = v 2 ( B v + D ) ( v ( B u + C v ) + D u ) 2
f v = B u v 2 + 2 D u v ( B u v + C v 2 + D u ) 2
x n + 1 = x n B x n x n 1 + C x n 1 2 + D x n x ¯ = 4 B + 4 C + D 2 D 2 ( B + C ) LAS for B> C 2 2 C D 2 D 2
a repeller for B< C 2 2 C D 2 D 2
a non-hyp. eq. for B= C 2 2 C D 2 D 2
no minimal period-two sol. f u = C v 2 ( v ( B u + C v ) + D u ) 2
f v = u ( B u + 2 C v ) ( B u v + C v 2 + D u ) 2