Skip to main content

Table 4 Equations of type (3,1)

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