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dc:title "HTML Summary of #4534 \n\nExponentially small splitting of invariant manifolds near a Hamiltonian-Hopf bifurcation\n\n";
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bibo:abstract "Consider an analytic two-degrees of freedom Hamiltonian system with an equilibrium\r\npoint that undergoes a Hamiltonian-Hopf bifurcation, i.e., the eigenvalues of\r\nthe linearized system at the equilibrium change from complex ±β ±iα (α,β > 0) for\r\nε > 0 to pure imaginary ±iω1 and ±iω2 (ω1 ≠ ω2 ≠ 0) for ε < 0. At ε = 0 the\r\nequilibrium has a pair of doubled pure imaginary eigenvalues. Depending on the sign\r\nof a certain coefficient of the normal form there are two main bifurcation scenarios. In\r\none of these (the stable case), two dimensional stable and unstable manifolds of the\r\nequilibrium shrink and disappear as ε → 0+. At any order of the normal form the stable\r\nand unstable manifolds coincide and the invariant manifolds are indistinguishable using\r\nclassical perturbation theory. In particular, Melnikov’s method is not capable to evaluate\r\nthe splitting.\r\nIn this thesis we have addressed the problem of measuring the splitting of these\r\nmanifolds for small values of the bifurcation parameter ε. We have estimated the size\r\nof the splitting which depends on a singular way from the bifurcation parameter. In\r\norder to measure the splitting we have introduced an homoclinic invariant ωε which\r\nextends the Lazutkin’s homoclinic invariant defined for area-preserving maps. The main\r\nresult of this thesis is an asymptotic formula for the homoclinic invariant. Assuming\r\nreversibility, we have proved that there is a symmetric homoclinic orbit such that its\r\nhomoclinic invariant can be estimated as follows,\r\nωε = ±2e−πα/2β (ω0 + O(ε1−μ)).\r\nwhere μ > 0 is arbitrarily small and ω0 is known as the Stokes constant. This asymptotic\r\nformula implies that the splitting is exponentially small (with respect to ε). When ω0 ≠ 0\r\nthen the invariant manifolds intersect transversely. The Stokes constant ω0 is defined\r\nfor the Hamiltonian at the moment of bifurcation only. We also prove that it does not\r\nvanish identically. Finally, we apply our methods to study homoclinic solutions in the\r\nSwift-Hohenberg equation. Our results show the existence of multi-pulse homoclinic\r\nsolutions and a small scale chaos."^^xsd:string;
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dct:date "2010-10";
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skos:prefLabel "QA Mathematics"@en .
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foaf:name "Mathematics Institute, University of Warwick"^^xsd:string;
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foaf:name "University of Warwick"^^xsd:string;
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foaf:familyName "Gaivão"^^xsd:string;
foaf:givenName "José Pedro"^^xsd:string;
foaf:name "José Pedro Gaivão"^^xsd:string;
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