 Author
 Year
 2017
 Title
 Newton flows for elliptic functions II
 Subtitle
 Structural stability: classification and representation
 Journal
 European Journal of Mathematics
 Volume  Issue number
 3  3
 Pages (fromto)
 691727
 Document type
 Article
 Faculty
 Faculty of Science (FNWI)
 Institute
 Kortewegde Vries Institute for Mathematics (KdVI)
 Abstract

In our previous paper we associated to each nonconstant elliptic function f on a torus T a dynamical system, the elliptic Newton flow corresponding to f. We characterized the functions for which these flows are structurally stable and showed a genericity result. In the present paper we focus on the classification and representation of these structurally stable flows. The phase portrait of a structurally stable elliptic Newton flow generates a connected, cellularly embedded graph G(f) on a torus T with r vertices, 2r edges and r faces that fulfil certain combinatorial properties (Euler, Hall) on some of its subgraphs. The graph G(f) determines the conjugacy class of the flow [classification]. A connected, cellularly embedded toroidal graph G with the above Euler and Hall properties, is called a Newton graph. Any Newton graph G can be realized as the graph G(f) of the structurally stable Newton flow for some function f. This leads to: up till conjugacy between flows and (topological) equivalency between graphs, there is a one to one correspondence between the structurally stable Newton flows and Newton graphs, both with respect to the same order r of the underlying functions f [representation]. Finally, we clarify the analogy between rational and elliptic Newton flows, and show that the detection of elliptic Newton flows is possible in polynomial time. The proofs of the above results rely on Peixoto’s characterization/classification theorems for structurally stable dynamical systems on compact 2dimensional manifolds, Stiemke’s theorem of the alternatives, Hall’s theorem of distinct representatives, the Heffter–Edmonds–Ringer rotation principle for embedded graphs, an existence theorem on gradient dynamical systems by Smale, and an interpretation of Newton flows as steady streams.
 URL
 go to publisher's site
 Other links
 Link to publication in Scopus
 Language
 English
 Permalink
 http://hdl.handle.net/11245.1/5f86120683854ee4be987a9211e854d3
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