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, DK2800 Kgs. Lyngby, Denmark)

Supervisor

Branner, Bodil (Department of Mathematics, Technical University of Denmark, DTU, DK2800 Kgs. Lyngby, Denmark)

Institution 
Technical University of Denmark, DTU, DK2800 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 nongeneric 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: 20061230
Update date: 20070224
Source: dtu
ID: 193586
Original MXD
