<rioxx xmlns="http://docs.rioxx.net/schema/v1.0/" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:dcterms="http://purl.org/dc/terms/" xmlns:rioxxterms="http://docs.rioxx.net/schema/v1.0/rioxxterms/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://docs.rioxx.net/schema/v1.0/ http://docs.rioxx.net/schema/v1.0/rioxx.xsd">
  <dc:creator>Gaivão, José Pedro</dc:creator>
  <dc:identifier>http://wrap.warwick.ac.uk/4534/1/WRAP_THESIS_Gaivao_2010.pdf</dc:identifier>
  <dc:format>application/pdf</dc:format>
  <dc:language>eng</dc:language>
  <dc:title>Exponentially small splitting of invariant manifolds near a Hamiltonian-Hopf bifurcation</dc:title>
  <dcterms:issued>2010-10</dcterms:issued>
  <dc:description>Consider an analytic two-degrees of freedom Hamiltonian system with an equilibrium&#13;
point that undergoes a Hamiltonian-Hopf bifurcation, i.e., the eigenvalues of&#13;
the linearized system at the equilibrium change from complex ±β ±iα (α,β &gt; 0) for&#13;
ε &gt; 0 to pure imaginary ±iω1 and ±iω2 (ω1 ≠ ω2 ≠ 0) for ε &lt; 0. At ε = 0 the&#13;
equilibrium has a pair of doubled pure imaginary eigenvalues. Depending on the sign&#13;
of a certain coefficient of the normal form there are two main bifurcation scenarios. In&#13;
one of these (the stable case), two dimensional stable and unstable manifolds of the&#13;
equilibrium shrink and disappear as ε → 0+. At any order of the normal form the stable&#13;
and unstable manifolds coincide and the invariant manifolds are indistinguishable using&#13;
classical perturbation theory. In particular, Melnikov’s method is not capable to evaluate&#13;
the splitting.&#13;
In this thesis we have addressed the problem of measuring the splitting of these&#13;
manifolds for small values of the bifurcation parameter ε. We have estimated the size&#13;
of the splitting which depends on a singular way from the bifurcation parameter. In&#13;
order to measure the splitting we have introduced an homoclinic invariant ωε which&#13;
extends the Lazutkin’s homoclinic invariant defined for area-preserving maps. The main&#13;
result of this thesis is an asymptotic formula for the homoclinic invariant. Assuming&#13;
reversibility, we have proved that there is a symmetric homoclinic orbit such that its&#13;
homoclinic invariant can be estimated as follows,&#13;
ωε = ±2e−πα/2β (ω0 + O(ε1−μ)).&#13;
where μ &gt; 0 is arbitrarily small and ω0 is known as the Stokes constant. This asymptotic&#13;
formula implies that the splitting is exponentially small (with respect to ε). When ω0 ≠ 0&#13;
then the invariant manifolds intersect transversely. The Stokes constant ω0 is defined&#13;
for the Hamiltonian at the moment of bifurcation only. We also prove that it does not&#13;
vanish identically. Finally, we apply our methods to study homoclinic solutions in the&#13;
Swift-Hohenberg equation. Our results show the existence of multi-pulse homoclinic&#13;
solutions and a small scale chaos.</dc:description>
  <dc:subject>QA</dc:subject>
  <dc:type>thesis</dc:type>
  <dc:relation>http://webcat.warwick.ac.uk/record=b2482741~S15</dc:relation>
</rioxx>