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 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.
Highlights
1 Motivation; recapitulation of earlier results In order to clarify the context of the present paper, we recapitulate some earlier results
Can be represented by the set of all ordered pairs ({[a1], . . . , [ar ]}, {[b1], . . . , [br ]}) of congruency classes mod that fulfil (2). This representation space can be endowed with a topology, say τ0, induced by the Euclidean topology on C, that is natural in the following sense: Given an elliptic function f of order r and ε > 0 sufficiently small, a τ0-neighborhood O of f exists such that for any g in O, the zeros for g are contained in ε-neighborhoods of the zeros for f
Conjugate flows are considered as equal, since we focus on qualitative aspects of the phase portraits
Summary
Let f be an elliptic (i.e., meromorphic, doubly periodic) function of order r ( 2) on the complex plane C with (ω1, ω2), Im ω2/ω1 > 0, as basic periods spanning a lattice (= ω1,ω2 ). The saddle in the case of a simple critical point (i.e., f does not vanish) is orthogonal and the two unstable (stable) separatrices constitute the “local” unstable (stable) manifold at this saddle Functions such as f correspond to meromorphic functions on the torus T ( ) (= C/ ω1,ω2 ). It is well known that the function f has precisely r zeros and r poles (counted by multiplicity) on the half open / half closed period parallelogram Pω1,ω2 given by t1ω1 + t2ω2 : 0 t1 < 1, 0 t2 < 1. Denoting these zeros and poles by a1, . Any pair ({a1, . . . , ar }, {b1, . . . , br }) that fulfils (1) determines (up to a multiplicative constant) an elliptic function with {a1, . . . , ar } and {b1, . . . , br } as zeros resp. poles in P (= Pω1,ω2 )
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
