Table 4 Equations of type $\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{1}\mathbf{)}$

$x_{n+1}=\frac{\beta x_n{x}_{n-1}+\gamma x_{n-1}^2+\delta x_n}{x_{n-1}{x}_n}$

$\overline{x}=\frac{\sqrt{{(\beta +\gamma )}^2+4\delta }+\beta +\gamma }{2}$

LAS for β>γ or $\beta \le \gamma \wedge {\beta }^2+2\beta \gamma +4\delta >3{\gamma }^2$

a saddle point for $\beta <\gamma \wedge {\beta }^2+2\beta \gamma +4\delta <3{\gamma }^2$

a non-hyp. eq. for $\beta <\gamma \wedge {\beta }^2+2\beta \gamma +4\delta =3{\gamma }^2$

no minimal period-two sol.

$f_u^{\prime }=-\frac{v\gamma }{u^2}$

$f_v^{\prime }=\frac{v^2\gamma -u\delta }{u{v}^2}$

$x_{n+1}=\frac{\beta x_n{x}_{n-1}+\gamma x_{n-1}^2+\delta x_n}{x_{n-1}^2}$

$\overline{x}=\frac{\sqrt{{(\beta +\gamma )}^2+4\delta }+\beta +\gamma }{2}$

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

$f_u^{\prime }=\frac{v\beta +\delta }{v^2}$

$f_v^{\prime }=\frac{-uv\beta -2u\delta }{v^3}$

$x_{n+1}=\frac{\beta x_n{x}_{n-1}+\gamma x_{n-1}^2+\delta x_n}{x_n}$

$\overline{x}=\frac{\delta }{1-\beta -\gamma }$

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

$\varphi =\frac{\delta (\gamma -\beta +1+\sqrt{(\beta -\gamma -1)(\beta +3\gamma -1)})}{2\gamma (-\beta +\gamma +1)}$

$\psi =\frac{\delta (\gamma -\beta +1-\sqrt{(\beta -\gamma -1)(\beta +3\gamma -1)})}{2\gamma (-\beta +\gamma +1)}$

a saddle point for β + 3γ<1

$f_u^{\prime }=-\frac{v^2\gamma }{u^2}$

$f_v^{\prime }=\frac{u\beta +2v\gamma }{u}$