Skip to main content

Table 2 Equations of type (2,1)

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 x n 1 + γ x n 1 2 x n x ¯ =t, t>0 for β + γ = 1
no eq. point for β + γ ≠ 1
a non-hyp. for β + γ = 1 no minimal period-two solution f u = v 2 γ u 2
f v = 2 v γ u +β
x n + 1 = β x n 1 x n + δ x n x n 1 2 x ¯ = 1 2 ( β 2 + 4 δ +β) a repeller for β,δ>0 no minimal period-two solution f u = v β + δ v 2
f v = u ( v β + 2 δ ) v 3
x n + 1 = γ x n 1 2 + δ x n x n 1 x n x ¯ = 1 2 ( γ 2 + 4 δ +γ) LAS for 4δ>3 γ 2
a saddle point for 4δ<3 γ 2
a non-hyp. for 4δ=3 γ 2
{ γ δ + δ 2 ( 4 δ 3 γ 2 ) 2 ( γ 2 δ ) , γ δ δ 2 ( 4 δ 3 γ 2 ) 2 ( γ 2 δ ) }
exists for 3 γ 2 <4δ<4 γ 2
a saddle point for 3 γ 2 <4δ<4 γ 2
f u = v γ u 2
f v = v 2 γ u δ u v 2
x n + 1 = γ x n 1 2 + δ x n x n 1 2 x ¯ = 1 2 ( γ 2 + 4 δ +γ) LAS for δ<2 γ 2
a repeller for δ>2 γ 2
a non-hyp. for δ=2 γ 2
possible Naimark-Sacker bifurcation f u = δ v 2
f v = 2 u δ v 3
x n + 1 = γ x n 1 2 + δ x n x n x ¯ = δ γ 1 for γ<1 LAS for 3γ<1
a saddle point for 3γ>1
a non-hyp. for 3γ = 1
{ϕ,ψ} for 3γ<1
ϕ= ( γ + 1 ) ( 1 3 γ ) δ 2 + γ δ + δ 2 γ ( γ + 1 )
ψ= ( γ + 1 ) ( 1 3 γ ) δ 2 + γ δ + δ 2 γ ( γ + 1 )
a saddle point for 3γ<1
f u = v 2 γ u 2
f v = 2 v γ u