Table 2 Equations of type $\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{1}\mathbf{)}$

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

$\overline{x}=t$, 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

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

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

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

$\overline{x}=\frac{1}{2}(\sqrt{{\beta }^2+4\delta }+\beta )$

a repeller for β,δ>0

no minimal period-two solution

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

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

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

$\overline{x}=\frac{1}{2}(\sqrt{{\gamma }^2+4\delta }+\gamma )$

LAS for $4\delta >3{\gamma }^2$

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

a non-hyp. for $4\delta =3{\gamma }^2$

{$\frac{\gamma \delta +\sqrt{{\delta }^2(4\delta -3{\gamma }^2)}}{2({\gamma }^2-\delta )}$, $\frac{\gamma \delta -\sqrt{{\delta }^2(4\delta -3{\gamma }^2)}}{2({\gamma }^2-\delta )}$ }

exists for $3{\gamma }^2<4\delta <4{\gamma }^2$

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

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

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

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

$\overline{x}=\frac{1}{2}(\sqrt{{\gamma }^2+4\delta }+\gamma )$

LAS for $\delta <2{\gamma }^2$

a repeller for $\delta >2{\gamma }^2$

a non-hyp. for $\delta =2{\gamma }^2$

possible Naimark-Sacker bifurcation

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

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

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

$\overline{x}=\frac{\delta }{\gamma -1}$ for γ<1

LAS for 3γ<1

a saddle point for 3γ>1

a non-hyp. for 3γ = 1

{ϕ,ψ} for 3γ<1

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

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

a saddle point for 3γ<1

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

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