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Exponentially small splitting of invariant manifolds near a HamiltonianHopf bifurcation
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Gaivão, José Pedro (2010) Exponentially small splitting of invariant manifolds near a HamiltonianHopf bifurcation. PhD thesis, University of Warwick.

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Official URL: http://webcat.warwick.ac.uk/record=b2482741~S15
Abstract
Consider an analytic twodegrees of freedom Hamiltonian system with an equilibrium
point that undergoes a HamiltonianHopf bifurcation, i.e., the eigenvalues of
the linearized system at the equilibrium change from complex ±β ±iα (α,β > 0) for
ε > 0 to pure imaginary ±iω1 and ±iω2 (ω1 ≠ ω2 ≠ 0) for ε < 0. At ε = 0 the
equilibrium has a pair of doubled pure imaginary eigenvalues. Depending on the sign
of a certain coefficient of the normal form there are two main bifurcation scenarios. In
one of these (the stable case), two dimensional stable and unstable manifolds of the
equilibrium shrink and disappear as ε → 0+. At any order of the normal form the stable
and unstable manifolds coincide and the invariant manifolds are indistinguishable using
classical perturbation theory. In particular, Melnikov’s method is not capable to evaluate
the splitting.
In this thesis we have addressed the problem of measuring the splitting of these
manifolds for small values of the bifurcation parameter ε. We have estimated the size
of the splitting which depends on a singular way from the bifurcation parameter. In
order to measure the splitting we have introduced an homoclinic invariant ωε which
extends the Lazutkin’s homoclinic invariant defined for areapreserving maps. The main
result of this thesis is an asymptotic formula for the homoclinic invariant. Assuming
reversibility, we have proved that there is a symmetric homoclinic orbit such that its
homoclinic invariant can be estimated as follows,
ωε = ±2e−πα/2β (ω0 + O(ε1−μ)).
where μ > 0 is arbitrarily small and ω0 is known as the Stokes constant. This asymptotic
formula implies that the splitting is exponentially small (with respect to ε). When ω0 ≠ 0
then the invariant manifolds intersect transversely. The Stokes constant ω0 is defined
for the Hamiltonian at the moment of bifurcation only. We also prove that it does not
vanish identically. Finally, we apply our methods to study homoclinic solutions in the
SwiftHohenberg equation. Our results show the existence of multipulse homoclinic
solutions and a small scale chaos.
Item Type:  Thesis or Dissertation (PhD)  

Subjects:  Q Science > QA Mathematics  
Library of Congress Subject Headings (LCSH):  Hamiltonian systems, Bifurcation theory, Manifolds (Mathematics)  
Official Date:  October 2010  
Dates: 


Institution:  University of Warwick  
Theses Department:  Mathematics Institute  
Thesis Type:  PhD  
Publication Status:  Unpublished  
Supervisor(s)/Advisor:  Gelfreich, Vassili  
Sponsors:  Fundação para a Ciência e a Tecnologia (FCT) (SFRH/BD/30596/2006)  
Extent:  x, 206 leaves : ill., charts  
Language:  eng 
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