# Table 5 Equations of type$(1,3)$

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. $f u ′ = C v 2 ( v ( B u + C v ) + D u ) 2$
$f v ′ = − u ( B u + 2 C v ) ( B u v + C v 2 + D u ) 2$