ABSTRACT
In this chapter we present all known periodic trichotomies of the third-order rational difference equation
xn+1 = α+ βxn + γxn−1 + δxn−2 A+Bxn + Cxn−1 +Dxn−2
, n = 0, 1, . . . (4.0.1)
with nonnegative parameters α, β, γ, δ, A,B,C,D and with arbitrary nonnegative initial conditions x−2, x−1, x0, such that the denominator is always positive. In Section 4.1 we present necessary and sufficient conditions for the exis-
tence of prime period-two solutions of Eq.(4.0.1). See also Section 5.9. In Section 4.2 we present the period-two trichotomies known for the rational
equations
xn+1 = α+ βxn + γxn−1
A+ xn , n = 0, 1, . . . , (4.0.2)
xn+1 = α+ γxn−1 + δxn−2
A+ xn−2 , n = 0, 1, . . . , (4.0.3)
and xn+1 =
α+ γxn−1 A+Bxn + xn−2
, n = 0, 1, . . . . (4.0.4)
Note that in addition to these three nonlinear period-two trichotomies, Eq.(4.0.1) contains a trivial period-two trichotomy for the linear equation
xn+1 = γ
A xn−1, n = 0, 1, . . . .
