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Mathematics 2016, 4(1), 20; doi:10.3390/math4010020

Birkhoff Normal Forms, KAM Theory and Time Reversal Symmetry for Certain Rational Map

1
Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USA
2
Department of Mathematics, University of Sarajevo, 71000 Sarajevo, Bosnia and Herzegovina
*
Author to whom correspondence should be addressed.
Academic Editor: Palle E.T. Jorgensen
Received: 6 February 2016 / Accepted: 14 March 2016 / Published: 18 March 2016
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Abstract

By using the KAM(Kolmogorov-Arnold-Moser) theory and time reversal symmetries, we investigate the stability of the equilibrium solutions of the system: x n + 1 = 1 y n , y n + 1 = β x n 1 + y n , n = 0 , 1 , 2 , , where the parameter β > 0 , and initial conditions x 0 and y 0 are positive numbers. We obtain the Birkhoff normal form for this system and prove the existence of periodic points with arbitrarily large periods in every neighborhood of the unique positive equilibrium. We use invariants to find a Lyapunov function and Morse’s lemma to prove closedness of invariants. We also use the time reversal symmetry method to effectively find some feasible periods and the corresponding periodic orbits. View Full-Text
Keywords: area preserving map; Birkhoff normal form; difference equation; KAM theory; periodic solutions; symmetry; time reversal area preserving map; Birkhoff normal form; difference equation; KAM theory; periodic solutions; symmetry; time reversal
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Denette, E.; Kulenović, M.R.S.; Pilav, E. Birkhoff Normal Forms, KAM Theory and Time Reversal Symmetry for Certain Rational Map. Mathematics 2016, 4, 20.

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