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

: 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 2 n − t n − 1 , n = 0, 1, 2, . . . , where are t − 1 , t 0 , α ∈ R , α (cid:54) = 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.


Introduction and Preliminaries
In this paper, we investigate the behavior of the polynomial quadratic second order difference equation t n+1 = αt n + βt 2 n − t n−1 , n = 0, 1, 2, . . . , (1) where is parameter β > 0, parameter α = 0 and the initial conditions t −1 , t 0 are real numbers. In References [1,2] the global dynamics of polynomial difference equations and are studied, where c > 0, d > 0, ζ ij ≥ 0, x −1 ≥ 0, x 0 ≥ 0 n = 0, 1, 2 . . . . (4) This difference equation was the subject of many mathematical results published in papers [3][4][5]. One of open problems in mathematics known as the 16th Hilbert problem is to find the upper limit number of periodic solutions of the system of polynomial differential equations of the forms dx dt = P(x, y) and dy dt = Q(x, y) depending on the degree and coefficients of the polynomials P(x, y) and Q(x, y), and then determine the local character of periodic solutions. Y. Ilyashenko andJ. Écalle (1991/1992) have shown that this number is finite in the case where degP = degQ = 2 (see [6]). Equation (1) is a special case of the equation which is equivalent to the following system of difference equations 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. One of the most famous examples of a polynomial difference equation is the second-order Hénon difference equation x n+1 = 1 + Bx n−1 − Ax 2 n , n = 0, 1, 2, ... .
The dynamics of this equation are very complex, for which the existence of a chaotic attractor has been proven ( [8]). Note that after the substitution Equation (1) becomes For α = 0 and α = 4 we get we get Hénon's mapping u n+1 = 1 − au 2 n + bu n−1 .
Assume that β > 0. Then Equation (1) has a unique equilibrium t = 0 for α = 2 and two equilibrium points t 1 = 0 and t 2 = 2−α β , where t 2 is positive for α < 2 and t 2 is negative for α > 2. Equation (1) can be transformed into the system The map T associated to System (8) takes the form Jacobian matrix of map T at the point (x, y) is given by Since det J T (x, y) = 1, the map T is an area preserving map.

Remark 1.
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.
This paper is organized as follows. By using KAM theorem, in Section 2, we will prove that the 9 2 , 5 , β > 0 are stable. In Section 3, we will prove that there exists an infinite number of periodic solutions. Also, we will study the minimal possible period (M.P.P.) for periodic orbit in the neighborhood of the By using the symmetries, in Section 4, we will find the periodic solutions with some periods and several numerical examples are given.
Proof. For the negative equilibrium t 2 = 2−α β , α ∈ (2, 6), we use the substitution and Equation (1) can be transformed into the system The map G associated with System (14) takes the form We see that and det J G (x, y) = 1. So, the map G is an area preserving map and we will apply KAM theory to System (14).
Lemma 3. M.P.P. for a periodic orbit in the neighborhood of the elliptic equilibrium (0, 0) is 3.

Symmetries
It is well known that symmetries play an important role because they have special dynamic behavior. See References [12,28] and Theorem 13 in Reference [13]. Now, the map T associated with the system (8) can be rewritten as follows : T(x, y) = (βx 2 + αx − y, x),
The involution R(x, y) = (y, x) is a reversor for the map T. Indeed, Thus T = I 1 • I 0 where I 0 (x, y) = R(x, y) = (y, x) and The symmetry lines corresponding to I 0 and I 1 are Periodic orbits on the symmetry line S 0 with even period n are searched for by starting with points (x 0 , x 0 ) ∈ S 0 and imposing that (x n/2 , y n/2 ) ∈ S 0 , where (x n/2 , y n/2 ) = T n/2 (x 0 , x 0 ).

Conclusions
By using KAM theory, under certain conditions on parameter α, we proved the stability of the zero equilibrium and the negative equilibrium of the difference Equation (1). Also, by using symmetries we proved the existence of periodic solutions with certain periods.
Author Contributions: Both authors have contributed equally and significantly to the contents of this paper.