Kolmogorov-Arnold-Moser Theory and Symmetries for a Polynomial Quadratic Second Order Difference Equation

Abstract

By using the Kolmogorov-Arnold-Moser (KAM) theory, we investigate the stability of two elliptic equilibrium points (zero equilibrium and negative equilibrium) of the difference equation t n+1 = αt n + βt 2 n − t n−1 , n = 0, 1, 2, . . . , where are t −1 , t 0 , α ∈ R, α = 0, β > 0. By using the symmetries we find the periodic solutions with some periods. Finally, some numerical examples are given to verify our theoretical results. MSC: 39A10; 39A11; 37E40; 37J40; 37N25In studying the global dynamics of (1) and (2), with non-negative initial conditions and non-negative parameters, the authors used the theory of monotonic maps.There is extensive literature on polynomial difference equations in the complex domain in the last 30 years by mathematicians. First investigations on polynomial difference equations with non-negative parameters and initial conditions were for a special case