Qualitative Behaviour of Newton Flows for Weierstrass Elliptic-Functions

摘要

We study the continuous, desingularized Newton method for Weierstrass elliptic-functions. This leads to a family of autonomous differential equations in the plane, which depends on two complex parameters omega(sub 1) and omega(sub 2). For the associated flows there are, up to conjugacy, precisely three possibilities. These are determined by the form of the parallelogram spanned by omega(sub 1) and omega(sub 2): square, rectangular but not square, and non-rectangular.