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Title: The role of global invariant manifolds of vector fields at homoclinic bifurcations
Author: Aguirre, Pablo
ISNI:       0000 0004 2725 4828
Awarding Body: University of Bristol
Current Institution: University of Bristol
Date of Award: 2012
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We consider certain kinds of homoclinic bifurcations in three-dimensional vector fields. These global bifurcations are characterized by the existence of a homo clinic orbit that converges to a saddle equilibrium in both forward and backward time. If the equilibrium has a complex pair of (stable) eigenvalues, it is a saddle-focus, and one speaks of a Shilnikov homoclinic orbit. In this case, the homoclinic orbit converges towards the equilibrium in a spiralling fashion. On the other hand, if the saddle equilibrium has two real (stable) eigenvalues, then the homoclinic orbit converges generically to the saddle along the direction given by the weak stable eigenvector. The possible unfoldings of a codimension-one homoclinic bifurcation depend on the sign of the saddle quantity: when it is negative, breaking the homoclinic orbit results in a single stable periodic orbit from a saddle-focus homoclinic orbit; one speaks of a simple Shilnikov bifurcation. However, when the saddle quantity is positive, then the mere existence of a Shilnikov homoclinic orbit induces complicated dynamics, and one speaks of a chaotic Shilnikov bifurcation. For a homoclinic orbit to a real saddle, on the other hand, always a single periodic orbit bifurcates, which is attracting when the saddle quantity is negative and of saddle type when it is positive. In this thesis we show how the global three-dimensional phase space is organized near certain homoclinic bifurcations by the two-dimensional global stable manifolds of equilibria and periodic orbits. To this end, we consider a model of a laser with optical injection that contains Shilnikov homoclinic orbits and a model by Sandstede that features different kinds of homoclinic bifurca- tions to a saddle. We find that, in the simple Shilnikov case, the stable manifold ofthe saddle-focus forms the basin boundary of the bifurcating stable periodic orbit. On the other hand, in the chaotic case, the stable manifold of the equilibrium is the accessible set of a chaotic saddle that contains countably many periodic orbits of saddle type. In the case of a homoclinic bifurcation to a saddle, the stable manifold of the saddle is either an orientable or nonorientable two-dimensional surface. A change of orientability occurs at two kinds of codimension-two homoclinic bifurcations, called inclination flip and orbit flip bifurcations. At either of these flip bifurcation points, the stable manifold is neither orientable nor nonorientable, but just at the transition between both states. We show how this transition occurs for the case of negative saddle quantity, and how the basin of attraction of the stable periodic orbit is organized in different ways by the stable manifold of the saddle depending on the (non)orientability of the bifurcation. Finally, we show how the stable manifold rearranges both itself and the overall dynamics in phase space near the codimension-two transition from a saddle to saddle-focus homoclinic bifurcation that occurs at a so-called Belyakov point.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available