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EXPONENTIALLY SMALL ASYMPTOTIC ESTIMATES FOR THE SPLITTING OF SEPARATRICES TO WHISKERED TORI WITH QUADRATIC AND CUBIC FREQUENCIES.

EXPONENTIALLY SMALL ASYMPTOTIC ESTIMATES FOR THE SPLITTING OF SEPARATRICES TO WHISKERED TORI WITH QUADRATIC AND CUBIC FREQUENCIES.

Authors :
DELSHAMS, AMADEU
GONCHENKO, MARINA
GUTIÉRREZ, PERE
Source :
Electronic Research Announcements in Mathematical Sciences. 2014, Vol. 21, p41-61. 21p.
Publication Year :
2014

Abstract

We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector ω = (1,Ω), where Ω is a quadratic irrational number, or a 3-dimensional torus with a frequency vector ω = (1,Ω,Ω2), where Ω is a cubic irrational number. Applying the Poincaré–Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show that such estimates depend strongly on the arithmetic properties of the frequencies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfilled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which is the so-called cubic golden number (the real root of x3 +x–1 = 0), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubic cases. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
19359179
Volume :
21
Database :
Academic Search Index
Journal :
Electronic Research Announcements in Mathematical Sciences
Publication Type :
Academic Journal
Accession number :
96805928
Full Text :
https://doi.org/10.3934/era.2014.21.41