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

Authors

Erin Denette, University of Rhode IslandFollow
Mustafa Kulenovic, University of Rhode IslandFollow
Esmir Pilav

Document Type

Article

Date of Original Version

3-18-2016

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 = βx_n/1+y_n, n = 0, 1, 2,…, where the parameter β > 0, and initial conditions x₀ and y₀ 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.

Citation/Publisher Attribution

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.

Available at: http://dx.doi.org/10.3390/math4010020

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.