# Table 2 Equations of type$(2,1)$

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
$ϕ= − ( γ + 1 ) ( 1 − 3 γ ) δ 2 + γ δ + δ 2 γ ( γ + 1 )$
$ψ= ( γ + 1 ) ( 1 − 3 γ ) δ 2 + γ δ + δ 2 γ ( γ + 1 )$
$f u ′ =− v 2 γ u 2$
$f v ′ = 2 v γ u$