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Table 3 Equations of type (2,2)

From: Local dynamics and global attractivity of a certain second-order quadratic fractional difference equation

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>1δ< 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(B1) 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