# Newton flows for elliptic functions IV, Pseudo Newton graphs: bifurcation & creation of flows

## TL;DR

This paper explores the bifurcation and creation of elliptic Newton flows, focusing on non-structurally stable flows represented by pseudo Newton graphs, providing insights into their stability and transition to stable configurations.

## Contribution

It introduces pseudo Newton graphs to analyze non-structurally stable elliptic Newton flows and investigates their bifurcation and creation, extending previous classifications of stable flows.

## Key findings

- Pseudo Newton graphs characterize non-stable flows.
- Bifurcation mechanisms for elliptic Newton flows are identified.
- Insights into the transition from non-stable to stable flows are provided.

## Abstract

An elliptic Newton flow is a dynamical system that can be interpreted as a continuous version of Newton's iteration method for finding the zeros of an elliptic function f. Previous work focusses on structurally stable flows (i.e., the phase portraits are topologically invariant under perturbations of the poles and zeros for f), including a classification / representation result for such flows in terms of Newton graphs (i.e., cellularly embedded toroidal graphs fulfilling certain combinatorial properties). The present paper deals with non-structurally stable elliptic Newton flows determined by pseudo Newton graphs (i.e., cellularly embedded toroidal graphs, either generated by a Newton graph, or the so called nuclear Newton graph, exhibiting only one vertex and two edges). Our study results into a deeper insight in the creation of structurally stable Newton flows and the bifurcation of non-structurally stable Newton flows.