# Table 3 Equations of type$(2,2)$

Equation Equilibrium point Stability of equilibrium point Period-two solution and stability Partial derivatives
$x n + 1 = β x n x n − 1 + γ x n − 1 2 C x n − 1 2 + x n$ $x ¯ = β + γ − 1 C$ for β + γ>1
no eq. point for β + γ ≤ 1
LAS for (3β + γ>3γ ≤ 3)γ>3
a saddle point for β<1β + γ>13β + γ<3
a non-hyp. eq. for γ<33β + γ = 3
${ ϕ 1 , ψ 1 }={0,γ/C}$-LAS for β<1
saddle for β>1; non-hyp. for β = 1
${ ϕ 2 , ψ 2 }$ exists for β<1, 3β + γ>3
a saddle point for β<1, 3β + γ>3
$ψ 1 = ( γ + 1 − β ) + ( γ + 1 − β ) ( 3 β + γ − 3 ) 2 c$
$ϕ 2 = ( γ + 1 − β ) − ( γ + 1 − β ) ( 3 β + γ − 3 ) 2 c$
$f u ′ = v 2 ( C v β − γ ) ( C v 2 + u ) 2$
$f v ′ = u ( v ( 2 γ − C v β ) + u β ) ( C v 2 + u ) 2$
$x n + 1 = β x n x n − 1 + γ x n − 1 2 B x n − 1 x n + x n$ $x ¯ = β + γ − 1 B$ 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
$f u ′ =− v 2 γ u 2 ( B v + 1 )$
$f v ′ = v γ ( B v + 2 ) + u β u ( B v + 1 ) 2$
$x n + 1 = β x n − 1 x n + δ x n B x n x n − 1 + x n − 1 2$ $x ¯ = 4 B δ + β 2 + 4 δ + β 2 ( B + 1 )$ LAS for $B β 2 >δ$
a repeller for $B β 2 <δ$
a non-hyp. for $B β 2 =δ$
no minimal period-two solution $f u ′ = v β + δ ( B u + v ) 2$
$f v ′ = − u ( B u δ + v 2 β + 2 v δ ) v 2 ( B u + v ) 2$
$x n + 1 = β x n − 1 x n + δ x n C x n − 1 2 + x n$ $x ¯ = 4 C δ + ( β − 1 ) 2 + β − 1 2 C$ a LAS for <2D(1 + β)
a repeller for >2D(1 + β)
a non-hyp. eq. for  = 2D(1 + β)
no minimal period-two solution $f u ′ = C v 2 ( v β + δ ) ( C v 2 + u ) 2$
$f v ′ = u ( u β − C v ( v β + 2 δ ) ) ( C v 2 + u ) 2$
$x n + 1 = γ x n − 1 2 + δ x n B x n − 1 x n + x n − 1 2$ $x ¯ = 4 B δ + γ 2 + 4 δ + γ 2 ( B + 1 )$ LAS for $3 ( B − 1 ) γ 2 ( B + 2 ) 2 <4δ$ or $δ<2(2B+1) γ 2$
saddle point for $B>1∧δ< 3 ( B − 1 ) γ 2 4 ( B + 2 ) 2$
non-hyp. eq. for $(4b+2) γ 2 =δ$ or $4 ( B + 2 ) 2 δ=3(B−1) γ 2$
repeller for $δ>2(2B+1) γ 2$
possible Naimark-Sacker bifurcation $f u ′ = δ − B v γ ( B u + v ) 2$
$f v ′ = u ( B v 2 γ − 2 v δ − B u δ ) v 2 ( B u + v ) 2$
$x n + 1 = γ x n − 1 2 + δ x n B x n x n − 1 + x n$ $x ¯ = 4 B δ + ( 1 − γ ) 2 + γ − 1 2 B$ LAS for $δ> ( γ + 1 ) ( 3 γ − 1 ) 4 B$
a saddle point for $δ< ( γ + 1 ) ( 3 γ − 1 ) 4 B$
a non-hyperbolic eq. for $δ= ( γ + 1 ) ( 3 γ − 1 ) 4 B$
{ϕ,ψ} exists for $( γ + 1 ) ( 3 γ − 1 ) 4 B <δ< γ ( γ + 1 ) B$
$ϕ= δ ( − 4 B δ − γ ( 3 γ + 2 ) + 1 + γ + 1 ) 2 ( − B δ + γ 2 + γ )$
$ψ= δ ( 4 B δ − γ ( 3 γ + 2 ) + 1 + γ + 1 ) 2 ( − B δ + γ 2 + γ )$
a saddle point for $( γ + 1 ) ( 3 γ − 1 ) 4 B <δ< γ ( γ + 1 ) B$
$f u ′ =− v 2 γ u 2 ( B v + 1 )$
$f v ′ = v γ ( B v + 2 ) − B u δ u ( B v + 1 ) 2$
$x n + 1 = γ x n − 1 2 + δ x n C x n − 1 2 + x n$ $x ¯ = 4 C δ + ( 1 − γ ) 2 + γ − 1 2 C$ LAS $δ< ( γ + 2 ) ( 2 γ + 1 ) C$ and $( 3 − γ ) ( 3 γ − 1 ) 16 C <δ$
a repeller for $δ> ( γ + 2 ) ( 2 γ + 1 ) C$
a saddle for $( 3 − γ ) ( 3 γ − 1 ) 16 C >δ$
a non-hyp. eq. for $( 3 − γ ) ( 3 γ − 1 ) 16 C =δ$ or $δ= ( γ + 2 ) ( 2 γ + 1 ) C$
possible Naimark-Sacker bifurcation $f u ′ = v 2 ( C δ − γ ) ( C v 2 + u ) 2$
$f v ′ = 2 u v ( γ − C δ ) ( C v 2 + u ) 2$