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

Erin Denette,Mustafa Kulenović,Esmir Pilav

doi:10.3390/math4010020

Erin Denette, Mustafa Kulenović + Show 1 more

Open Access

https://doi.org/10.3390/math4010020

Copy DOI

Journal: Mathematics	Publication Date: Mar 18, 2016
Citations: 17	License type: CC BY 4.0

Affiliation: University of Sarajevo, University of Rhode Island

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.

Highlights

The following rational system of difference equations: x n +1 = yn y n +1 1+ y n, n = 0, 1, . . . (1)n = 0, 1, . . . , (2)and the corresponding equation: y n +1 =
The method of invariants for the construction of a Lyapunov function and proving stability of the equilibrium points was used successfully in [5,6,12], and the KAM theory was used for the same objective in [12,13,14,15,16]
The second section contains a derivation of the Birkhoff normal form for map T and an application of the KAM theory, which proves stability of the equilibrium and the existence of an infinite number of periodic solutions for β 6= 2

Summary

Introduction

The following rational system of difference equations:. , n = 0, 1,. The method of invariants for the construction of a Lyapunov function and proving stability of the equilibrium points was used successfully in [5,6,12], and the KAM theory was used for the same objective in [12,13,14,15,16]. In the case when a difference equation’s corresponding map is area preserving and does not possess an invariant, the only tool left seems to be KAM theory, see [17] for such an example. The second section contains a derivation of the Birkhoff normal form for map T and an application of the KAM theory, which proves stability of the equilibrium and the existence of an infinite number of periodic solutions for β 6= 2. We use time reversal symmetry method [13,20] based on the symmetries to effectively find some feasible periods and corresponding orbits of the map T

The KAM Theory and Birkhoff Normal Form

Invariant

Symmetries

Conclusions