Birkhoff Normal Forms, Kam Theory, Periodicity and Symmetries for Certain Rational Difference Equation With Cubic Terms

Abstract

By using the Kolmogorov-Arnold-Moser (KAM) theory, we investigate the stability of the positive elliptic equilibrium point of the difference equation xn+1 = Ax 3 n + B axn−1 , n = 0, 1, 2,. .. where the parameters A, B, a and the initial conditions x−1, x0 are positive numbers. The specific feature of this difference equation is the fact that we were not able to use the invariant to prove stability or to find feasible periods of the solutions.

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

“…. Now, we will find the Birkhoff normal form of System (8) (see Theorem 1 in Reference [21]). The change of variables…”

“…By using the transformation ξ n = r n + is n , η n = r n − is n , we obtain (see Theorem 1, for l = 4, in Reference [21]):…”

“…Namely, we cannot find the invariant of this equation and we can use only Kolmogorov-Arnold-Moser (KAM) theory to investigate the dynamics of this equation. Some results about using KAM theory are given in References [9][10][11][12][13][14][15][16][17][18][19][20][21]. Other techniques have been used (not only KAM theory) to study the Lyness' equation (see References [9][10][11]17,[22][23][24][25][26][27]).…”

“…The definition of an area preserving map is given in Reference [21] (Definition 1). Also, for the basic results about Birkhoff normal forms and the KAM theory, see References [8,20,21,[28][29][30][31] (especially see Theorems 1-4 and Remark 1 in [21]).…”

Global Dynamics of the System of Two Exponential Difference Equations Mai Nam Phong

“…EJMAA-2019/7 (2) In this case, the unique positive equilibrium point of the system (43) is unstable. Moreover, in Figure 3, the plot of x n is shown in Figure 3 (a), the plot of y n is shown in Figure 3 (b), and a phase portrait of the system (43) is shown in Figure 3 (c).…”