Beta 1


Title Bifurcations of Complex Polynomial Vector Fields
Author Dias, Kealey (Danish Center for Applied Mathematics and Mechanics, Department of Mathematics, Technical University of Denmark, DTU, DK-2800 Kgs. Lyngby, Denmark)
Supervisor Branner, Bodil (Department of Mathematics, Technical University of Denmark, DTU, DK-2800 Kgs. Lyngby, Denmark)
Institution Technical University of Denmark, DTU, DK-2800 Kgs. Lyngby, Denmark
Thesis level Master's thesis
Year 2006
Abstract Complex polynomial vector fields in the plane form a restricted subset of vector fields in the plane in general. We analyze these vector fields by reducing them to their fundamental parts (separatrices, equilibrium points and their basins) and defining out of these combinatorial and analytic invariants. With these invariants, one can build a rectified surface for the vector field. Reversely, we want to determine if this vector field is uniquely determined by its combinatorial invariant and its analytic invariants. This fundamental theorem is proven for generic vector fields as follows: from the combinatorial invariant and analytic invariants, we construct a Riemann surface homemorphic to the Riemann sphere and having an induced vector field, prove that the Riemann surface is isomorphic to the Riemann spheere, and finally prove the fundamental theorem. We describe non-generic vector fields for low degree, and extend the ideas of the fundamental theorem to this more general case. A description of some of the bifurcations that can occur for low degree are given. A proof of genericity for vector fields having simple equilibrium points at the d roots of unity, d odd is given. The subject of parameter spaces for low degree polynomial vector fields is touched on.
Pages 113
Admin Creation date: 2006-12-30    Update date: 2007-02-24    Source: dtu    ID: 193586    Original MXD