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 = α t n + β t n 2 − 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.
Highlights
We investigate the behavior of the polynomial quadratic second order difference equation tn+1 = αtn + βt2n − tn−1, n = 0, 1, 2, . . . , (1)
In 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
First investigations on polynomial difference equations with non-negative parameters and initial conditions were for a special case xn+1 = Bxn xn−1 + Exn−1 + F, n = 0, 1, 2
Summary
First investigations on polynomial difference equations with non-negative parameters and initial conditions were for a special case xn+1 = Bxn xn−1 + Exn−1 + F, n = 0, 1, 2,. Which is equivalent to the following system of difference equations i i− j j yn , This system is a special case of discrete version of the 16th Hilbert problem and for which, in Reference [7], the authors have shown that under certain conditions may have infinitely many periodic solutions of the period 2, which means that the discrete version of 16th Hilbert problem does not hold. When the equilibrium point of Equation (1) is a non-hyperbolic of elliptic type and T is an area-preserving map, we can apply KAM theory to the investigation of its stability. For the bifurcation diagrams (B.D.) which indicate the appearance of chaos see Figures 1–4
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