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Title: On homoclinic orbits to center manifolds in Hamiltonian systems
Author: Giles, William
ISNI:       0000 0004 5349 4833
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2015
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The objects of study in this thesis are Hamiltonian systems of ordinary differential equations possessing homoclinic orbits. A homoclinic orbit is a solution of the system which converges to the same invariant set as time approaches both positive and negative infinity. In our case, the invariant set in question is assumed to be a nonhyperbolic equilibrium state. Such an equilibrium state possesses a center manifold, containing all orbits which remain close to the equilibrium. We are concerned with finding orbits which converge to orbits in the center manifold in both time directions. We consider firstly the case in which the nonhyperbolic eigenvalues at the equilibrium consist of pairs of nonzero purely imaginary eigenvalues. We study the set of homoclinics to the center manifold by constructing an operator on a suitable function space whose zeros correspond to homoclinics. We use a Lyapunov-Schmidt technique to reduce the problem to that of studying the zero set of a real-valued function defined on the center manifold, which has a critical point at the origin. A formula is found for the Hessian matrix at this critical point, involving the so called scattering matrix. Under nonresonance and nondegeneracy conditions, we characterise the possible Morse indices of the Hessian, permitting an application of the Morse lemma to describe the set of homoclinics. We also consider special cases, including reversible systems. We then consider a more geometric approach to the problem, allowing us to define a nonlinear analogue of the scattering matrix using stable and unstable foliations of the invariant manifolds. We use this approach to unfold the system in parametrised families - we consider here also the case of a two dimensional center manifold corresponding to zero eigenvalues - bifurcation diagrams are produced for homoclinics to the origin in this case. The effects of additional reversible structure are again considered.
Supervisor: Lamb, Jeroen; Turaev, Dmitry Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral