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Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 1 Equations of type (1,2)

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 − 1 x n C x n − 1 2 + x n

x ¯ = β − 1 C for β>1

no eq. point for β ≤ 1

LAS for β>1

no period-two solution

f u ′ = C v 3 β ( C v 2 + u ) 2

f v ′ = u β ( u − C v 2 ) ( C v 2 + u ) 2

x n + 1 = γ x n − 1 2 C x n − 1 2 + x n

x ¯ = γ − 1 C for γ>1

no eq. point for γ ≤ 1

LAS for γ>3

a saddle point for 1<γ<3

a non-hyp. eq. for γ = 3

{0,γ/C}-LAS

{ γ + 1 − ( γ − 3 ) ( γ + 1 ) 2 C , γ + 1 + ( γ − 3 ) ( γ + 1 ) 2 C }

a saddle point for γ>3

a non-hyp. eq. for γ = 3

f u ′ =− γ v 2 ( C v 2 + u ) 2

f v ′ = 2 γ u v ( C v 2 + u ) 2

x n + 1 = γ x n − 1 2 B x n − 1 x n + x n

x ¯ = γ − 1 B for γ>1

no eq. point for γ ≤ 1

a saddle point for γ>1

no period-two solution

f u ′ =− γ v 2 u 2 ( B v + 1 )

f v ′ = γ v ( B v + 2 ) u ( B v + 1 ) 2

x n + 1 = δ x n B x n x n − 1 + x n − 1 2

x ¯ = δ B + 1

a repeller for δ>0, B>0

no period-two solution

f u ′ = δ ( B u + v ) 2

f v ′ =− u δ ( B u + 2 v ) ( B u v + v 2 ) 2

x n + 1 = δ x n C x n − 1 2 + x n

x ¯ = 4 C δ + 1 − 1 2 C

LAS for cδ<2

a repeller for cδ>2

a non-hyp. eq. for cδ = 2

no period-two solution

f u ′ = C v 2 δ ( C v 2 + u ) 2

f v ′ =− 2 C u v δ ( C v 2 + u ) 2

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