# Table 1 Equations of type$(1,2)$

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 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 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 $f u ′ = C v 2 δ ( C v 2 + u ) 2$
$f v ′ =− 2 C u v δ ( C v 2 + u ) 2$