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|.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:714974 |
| Date | January 2017 |
| Creators | Moutsinas, Giannis |
| Publisher | University of Warwick |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://wrap.warwick.ac.uk/88465/ |
Page generated in 0.0016 seconds