Table 3 Equations of type $\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{2}\mathbf{)}$

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

$\overline{x}=\frac{\beta +\gamma -1}{C}$ for β + γ>1

no eq. point for β + γ ≤ 1

LAS for (3β + γ>3∧γ ≤ 3)∨γ>3

a saddle point for β<1∧β + γ>1∧3β + γ<3

a non-hyp. eq. for γ<3∧3β + γ = 3

$\{{\varphi }_1,{\psi }_1\}=\{0,\gamma /C\}$-LAS for β<1

saddle for β>1; non-hyp. for β = 1

$\{{\varphi }_2,{\psi }_2\}$ exists for β<1, 3β + γ>3

a saddle point for β<1, 3β + γ>3

${\psi }_1=\frac{(\gamma +1-\beta )+\sqrt{(\gamma +1-\beta )(3\beta +\gamma -3)}}{2c}$

${\varphi }_2=\frac{(\gamma +1-\beta )-\sqrt{(\gamma +1-\beta )(3\beta +\gamma -3)}}{2c}$

$f_u^{\prime }=\frac{v^2(Cv\beta -\gamma )}{{(C{v}^2+u)}^2}$

$f_v^{\prime }=\frac{u(v(2\gamma -Cv\beta )+u\beta )}{{(C{v}^2+u)}^2}$

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

$\overline{x}=\frac{\beta +\gamma -1}{B}$ for β + γ>1

LAS for β>γ + 1

a saddle for 1 − γ<β<γ + 1

a non-hyp. for β = γ + 1

{ϕ,ψ} for ϕ,ψ>0

for β = γ + 1

$f_u^{\prime }=-\frac{v^2\gamma }{u^2(Bv+1)}$

$f_v^{\prime }=\frac{v\gamma (Bv+2)+u\beta }{u{(Bv+1)}^2}$

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

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

LAS for $B{\beta }^2>\delta$

a repeller for $B{\beta }^2<\delta$

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

no minimal period-two solution

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

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

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

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

a LAS for cδ<2D(1 + β)

a repeller for cδ>2D(1 + β)

a non-hyp. eq. for cδ = 2D(1 + β)

no minimal period-two solution

$f_u^{\prime }=\frac{C{v}^2(v\beta +\delta )}{{(C{v}^2+u)}^2}$

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

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

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

LAS for $\frac{3(B-1){\gamma }^2}{{(B+2)}^2}<4\delta$ or $\delta <2(2B+1){\gamma }^2$

saddle point for $B>1\wedge \delta <\frac{3(B-1){\gamma }^2}{4{(B+2)}^2}$

non-hyp. eq. for $(4b+2){\gamma }^2=\delta$ or $4{(B+2)}^2\delta =3(B-1){\gamma }^2$

repeller for $\delta >2(2B+1){\gamma }^2$

possible Naimark-Sacker bifurcation

$f_u^{\prime }=\frac{\delta -Bv\gamma }{{(Bu+v)}^2}$

$f_v^{\prime }=\frac{u(B{v}^2\gamma -2v\delta -Bu\delta )}{v^2{(Bu+v)}^2}$

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

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

LAS for $\delta >\frac{(\gamma +1)(3\gamma -1)}{4B}$

a saddle point for $\delta <\frac{(\gamma +1)(3\gamma -1)}{4B}$

a non-hyperbolic eq. for $\delta =\frac{(\gamma +1)(3\gamma -1)}{4B}$

{ϕ,ψ} exists for $\frac{(\gamma +1)(3\gamma -1)}{4B}<\delta <\frac{\gamma (\gamma +1)}{B}$

$\varphi =\frac{\delta (-\sqrt{4B\delta -\gamma (3\gamma +2)+1}+\gamma +1)}{2(-B\delta +{\gamma }^2+\gamma )}$

$\psi =\frac{\delta (\sqrt{4B\delta -\gamma (3\gamma +2)+1}+\gamma +1)}{2(-B\delta +{\gamma }^2+\gamma )}$

a saddle point for $\frac{(\gamma +1)(3\gamma -1)}{4B}<\delta <\frac{\gamma (\gamma +1)}{B}$

$f_u^{\prime }=-\frac{v^2\gamma }{u^2(Bv+1)}$

$f_v^{\prime }=\frac{v\gamma (Bv+2)-Bu\delta }{u{(Bv+1)}^2}$

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

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

LAS $\delta <\frac{(\gamma +2)(2\gamma +1)}{C}$ and $\frac{(3-\gamma )(3\gamma -1)}{16C}<\delta$

a repeller for $\delta >\frac{(\gamma +2)(2\gamma +1)}{C}$

a saddle for $\frac{(3-\gamma )(3\gamma -1)}{16C}>\delta$

a non-hyp. eq. for $\frac{(3-\gamma )(3\gamma -1)}{16C}=\delta$ or $\delta =\frac{(\gamma +2)(2\gamma +1)}{C}$

possible Naimark-Sacker bifurcation

$f_u^{\prime }=\frac{v^2(C\delta -\gamma )}{{(C{v}^2+u)}^2}$

$f_v^{\prime }=\frac{2uv(\gamma -C\delta )}{{(C{v}^2+u)}^2}$