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A geometric approach to evaluation-transversality techniques in generic bifurcation theory Aalto, Søren Karl


The study of bifurcations of vectorfields is concerned with changes in qualitative behaviour that can occur when a non-structurally stable vectorfield is perturbed. In a sense, this is the study of how such a vectorfield fails to be structurally stable. Finding a systematic approach to the study of such questions is a difficult problem. One approach to bifurcations of vectorfields, known as "generic bifurcation theory," is the subject of much of the work of Sotomayor (Sotomayor [1973a], Sotomayor [1973b],Sotomayor [1974]). This approach attempts to construct generic families of k-parameter vectorfields (usually for k=1), for which all the bifurcations can be described. In Sotomayor [1973a] it is stated that the vectorfields associated with the "generic" bifurcations of individual critical elements for k-parameter vectorfields form submanifolds of codimension ≤ k of the Banach space ϰʳ (M) of vectorfields on a compact manifold M. The bifurcations associated with one of these submanifolds of codimension-k are called generic codimension-k bifurcations. In Sotomayor [1974] the construction of these submanifolds and the description of the associated bifurcations of codimension-1 for vectorfields on two dimensional manifolds is presented in detail. The bifurcations that occur are due to the parameterised vectorfield crossing one of these manifolds transversely as the parameter changes. Abraham and Robbin used transversality results for evaluation maps to prove the Kupka-Smale theorem in Abraham and Robbin [1967]. In this thesis, we shall show how an extension of these evaluation transversality techniques will allow us to construct the submanifolds of ϰʳ (M) associated with one type of generic bifurcation of critical elements, and we shall consider how this approach might be extended to include the other well known generic bifurcations. For saddle-node type bifurcations of critical points, we will show that the changes in qualitative behaviour are related to geometric properties of the submanifold Σ₀ of ϰʳ (M) x M, where Σ₀ is the pull-back of the set of zero vectors-or zero section-by the evaluation map for vectorfields. We will look at the relationship between the Taylor series of a vector-field X at a critical point ⍴ and the geometry of Σ₀ at the corresponding point (X,⍴) of ϰʳ (M) x M. This will motivate the non-degeneracy conditions for the saddle-node bifurcations, and will provide a more general geometric picture of this approach to studying bifurcations of critical points. Finally, we shall consider how this approach might be generalised to include other bifurcations of critical elements.

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