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

Tarek F. Ibrahim and Zehra Nurkanović
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Tarek F. Ibrahim: Department of Mathematics, Faculty of Sciences and Arts in Mahayel Aser, King Khalid University, Abha, Sarat Abida 61914, Saudi Arabia
Zehra Nurkanović: Department of Mathematics, University of Tuzla, 75000 Tuzla, Bosnia and Herzegovina

Mathematics, 2019, vol. 7, issue 9, 1-16

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.

Keywords: area-preserving map; difference equation; stability; bifurcation; Birkhoff normal form; KAM theorem; twist coefficient; symmetry; periodic solutions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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