# Table 4 Equations of type$(3,1)$

Equation Equilibrium point Stability of equilibrium point Period-two solution and stability Partial derivatives
$x n + 1 = β x n x n − 1 + γ x n − 1 2 + δ x n x n − 1 x n$ $x ¯ = ( β + γ ) 2 + 4 δ + β + γ 2$ LAS for β>γ or $β≤γ∧ β 2 +2βγ+4δ>3 γ 2$
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
$f u ′ =− v 2 γ u 2$
$f v ′ = u β + 2 v γ u$