Newton flows are dynamical systems generated by a continuous, desingularized Newton method for mappings from a Euclidean space to itself. We focus on the special case of meromorphic functions on the complex plane. Inspired by the analogy between the rational (complex) and the elliptic (i.e., doubly periodic meromorphic) functions, a theory on the class of so-called Elliptic Newton flows is developed.
With respect to an appropriate topology on the set of all elliptic functions $f$ of fixed order $r$ ($\geq$ 2) we prove: For almost all functions $f$, the corresponding Newton flows are structurally stable i.e., topologically invariant under small perturbations of the zeros and poles for $f$ [genericity].
The phase portrait of a structurally stable elliptic Newton flow generates a connected, cellularly embedded, graph $G(f)$ on $T$ with $r$ vertices, 2$r$ 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 [characterization].
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 $f$ [classification].
This leads to: up till conjugacy between flows and(topological) equivalency between graphs, there is a 1-1 correspondence between the structurally stable Newton flows and Newton graphs, both with respect to the same order $r$ of the underlying functions $f$ [representation].
In particular, it follows that in case $r$ = 2, there is only one (up to conjugacy) structurally stabe elliptic Newton flow, whereas in case $r$ = 3, we find a list of nine graphs, determining all possibilities.
Moreover, we pay attention to the so-called nuclear Newton flows of order $r$, and indicate how - by a bifurcation procedure - any structurally stable elliptic Newton flow of order $r$ can be obtained from such a nuclear flow.
Finally, we show that the detection of elliptic Newton flows is possible in polynomial time.
The proofs of the above results rely on Peixoto's characterization/classication theorems for structurally stable dynamical systems on compact 2-dimensional manifolds, Stiemke's theorem of the alternatives, Hall's theorem of distinct representatives, the Heter-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.
|Name||Memorandum of the Department of Applied Mathematics|
- Newton graph (elliptic-
- Newton flow (rational-
- steady stream
- face traversal procedure
- elliptic function (Jacobian
- Dynamical system (gradient-)
- cellularly embedded toroidal (distinguished) graph
- structural stability