Moutsinas, Giannis (2017) Splitting of separatrices in area-preserving maps close to 1:3 resonance. PhD thesis, University of Warwick.

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Abstract

We consider a real analytic family of area-preserving maps on C2, fµ, depending analytically on the parameter, such that f0 is a map at 1:3 resonance. Such maps can be formally embedded in an one degree of freedom Hamiltonian system, called the normal form of the map. We denote the third iterate of the map by Fµ = f3µ.

We show that given a certain non degeneracy condition on the map F0, there exists a Stokes constant, θ, that when it does not vanish, it describes the splitting of the separatrices that the normal form predicts. We show that this constant can be approximated numerically for any non-degenerate map F0.

For a non-vanishing and small enough µ, we show that if the Stokes constant does not vanish the separatrices split. Moreover, let Ω be the area of the parallelogram defined by the 2 vectors tangent at the two separatrices at a homoclinic point. For any M ε N we have the estimate.

In this equation λµ is the largest eigenvalue of the saddle points around the origin and θn's are real constants with θ0 = 4π|θ|.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Differentiable mappings, Hamiltonian systems
Official Date:	January 2017
Dates:	Date Event January 2017 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Gelfreich, Vassili
Sponsors:	Engineering and Physical Sciences Research Council
Format of File:	pdf
Extent:	182 pages : illustrations, charts
Language:	eng

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