The Qualitative Behaviour of Newton Flows for Weierstrass’ ℘-Functions

Abstract

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

On the geometry, flows and visualization of singular complex analytic vector fields on Riemann surfaces

“…It should be noted from the outset that the method in question exploits a well-known characteristic of Newton vector fields, namely that their streamlines can be easily recognized by a geometrical argument (see Lemma 4). Yet, it is interesting to note that apparently this method is unknown (or at least not actively used), even for those who study Newton vector fields: for instance in [36], [71]. In particular, though they show that the Newton flow associated to the Weierstrass ℘-functions can be characterized/classified (up to conjugacy) into three types of behaviour, and that they actually show phase portraits of the Newton flow associated to Weierstrass ℘-function and to Jacobi's sn-function, they still use a traditional integration-based algorithm (4-th order Runge-Kutta) for the visualization of the vector field.…”

“…is visualized using the techniques described in this note: the strip flows of ρ(z) = arg (℘(z)) are plotted in Figure 19 (a), moreover the corresponding vector field on the torus is visualized in Figure 19 (b). It should be noted that the vector field given by ( 34) was previously studied by G. F. Helminck et al, see [36] for further details. In particular they showed that up to conjugation the family of vector fields of the form (34) consist of three classes characterized 11 by the form of the parallelogram spanned by the parameters ω 1 , ω 2 defining 12 ℘(z).…”

“…In particular they showed that up to conjugation the family of vector fields of the form (34) consist of three classes characterized 11 by the form of the parallelogram spanned by the parameters ω 1 , ω 2 defining 12 ℘(z). They visualized a couple of trajectories using a 4 th order Runge-Kutta integration algorithm for the case corresponding to ω 1 = 1, ω 2 = 1 4 + 5 4 i, compare Figure 19 (a) with [36] figure 8. In the case of Riemann surfaces of genus g > 1, a similar scheme will work: the Γ-invariant vector field X(z) given by (33) (which is characterized by the Γ-invariant function f (z) which is analytic on H\S, where S is a discrete set) will be meromorphic in C, and hence one will be able to explicitly represent them as Newton vector fields.…”

“…It can be proved that ℘ is doubly periodic with periods precisely ω 1 and ω 2 . See [36] and references therein for further details. 13 The Γ-invariant functions in this case are known as automorphic functions and have very interesting applications to number theory, amongst other things.…”

“…so that the G ij are constant on the trajectories of dx dτ = H(x). Note that in this case, as opposed to the complex case, one still has to solve a system of differential equations to find the auxiliary functions G ij , but these equations are usually much simpler to solve than the original ones that define the trajectories and that are solutions to (36). In a couple of cases examined in [62], [18] these auxiliary functions G ij can be found, but there is no known technique that works in general.…”