Birkhoff Normal Forms and KAM Theory for Gumowski-Mira Equation

Abstract

By using the KAM theory we investigate the stability of equilibrium solutions of the Gumowski-Mira equation: x n+1 = (2ax n)/(1 + x n 2) − x n−1, n = 0,1,…, where x −1, x 0 ∈ (−∞, ∞), and we obtain the Birkhoff normal forms for this equation for different equilibrium solutions.

Stability, periodicity, and symmetries of certain second‐order fractional difference equation with quadratic terms via KAM theory

“…(2) is very similar to Gumowski-Mira equation [2][3][4][5][6][7][8][9]. These equations were considered by either the Kolmogorov-Arnold-Moser (KAM) theory as in [8][9][10][11][12] or a combination of algebraic and geometric techniques as in [2-4, 13, 14]. The second technique was always based on the existence of invariants which analysis led to the properties of the solutions and in particular, to the results on feasible periods, chaotic solutions and so on.…”

Non-integrability of measure preserving maps via Lie symmetries

“…The Birkhoff constants given in the above lemma are also obtained in [19] to study the stability of the elliptic fixed points.…”

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

“…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]).…”