Newton flows for elliptic functions III & IV: Pseudo Newton graphs: bifurcation and creation of flows

Gerard F. Helminck*, Frank Twilt

*Corresponding author for this work

    Research output: Contribution to journalArticleAcademicpeer-review

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    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 focuses 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. As it requires the classification of all third order Newton graphs, we present this classification.

    Original languageEnglish
    Pages (from-to)1364-1395
    Number of pages32
    JournalEuropean Journal of Mathematics
    Issue number4
    Publication statusPublished - 1 Dec 2019


    • Angle property
    • Cellularly embedded toroidal (distinguished) graph
    • Desingularized elliptic Newton flow
    • Dynamical system
    • Elliptic function
    • Euler property
    • Face traversal procedure
    • Hall condition
    • Newton graph (elliptic-, nuclear-, pseudo-)
    • Phase portrait
    • Structural stability


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