Birkhoff Normal Forms, Kam Theory, Periodicity and Symmetries for Certain Rational Difference Equation With Cubic Terms
DOI:
https://doi.org/10.5644/SJM.12.2.08Keywords:
Area preserving map, Birkhoff normal form, KAM theorem, stability, twist coefficientAbstract
Dedicated to the memory of Professor Mahmut Bajraktarevi´c
By using the Kolmogorov-Arnold-Moser (KAM) theory, we investigate the stability of the positive elliptic equilibrium point of the difference equation $x_{n+1} = \frac{Ax_n^3 + B}{ax_{n-1}}, n=0,1,2,...$ where the parameters $A, B, a$ and the initial conditions $x_{-1},x_0$ 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.
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