2016
DOI: 10.3390/math4010020
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Birkhoff Normal Forms, KAM Theory and Time Reversal Symmetry for Certain Rational Map

Abstract: By using the KAM(Kolmogorov-Arnold-Moser) theory and time reversal symmetries, we investigate the stability of the equilibrium solutions of the system: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… Show more

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Cited by 4 publications
(2 citation statements)
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References 18 publications
(11 reference statements)
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“…As we can see in [3,4,5,7], symmetries are important in the study of area-preserving maps because they yield special dynamic behavior. A transformation R of the plane is a time reversal symmetry for T if R −1 • T • R = T −1 .…”
Section: Symmetriesmentioning
confidence: 99%
“…As we can see in [3,4,5,7], symmetries are important in the study of area-preserving maps because they yield special dynamic behavior. A transformation R of the plane is a time reversal symmetry for T if R −1 • T • R = T −1 .…”
Section: Symmetriesmentioning
confidence: 99%
“…See also [21] for the results on the stability of Lyness equation with period two coefficient by using KAM theory. In [1,7] authors consider the rational second-order difference equation…”
Section: Introductionmentioning
confidence: 99%