Existence and Local Stability of Prime Period-two Solutions of Certain Quadratic Rational Second Order Difference Equation

Abstract

In this paper we proved the existence and local stability of prime period-two solutions for the equation 𝐱𝐧􀬾𝟏 􀵌 𝛂𝐱𝐧 𝟐 􀬾𝛃𝐱𝐧􀬾𝛄𝐱𝐧􀰷𝟏 𝐀𝐱𝐧 𝟐 􀬾𝐁𝐱𝐧􀬾𝐂𝐱𝐧􀰷𝟏 , for certain values of parameters ,,,A,B,C0, where ++>0 , A+B+C>0, and where the initial conditions x₋₁, x₀>0 are arbitrary real numbers such that at least one is strictly positive. For the obtained periodic solutions, it is possible to be locally asymptotically stable, saddle points or nonhyperbolic points. The existence of repeller points is not possible.