On the classification of plane graphs representing structurally stable rational Newton flows

We study certain plane graphs, called Newton graphs, representing a special class of dynamical systems which are closely related to Newton's iteration method for finding zeros of (rational) functions defined on the complex plane. These Newton graphs are defined in terms of nonvanishing angles between edges at the same vertex. We derive necessary and sufficient conditions -of purely combinatorial nature- for an arbitrary plane graph in order to be topologically equivalent with a Newton graph. Finally, we analyse the structure of Newton graphs and prove the existence of a polynomial algorithm to recognize such graphs

Newton flows for elliptic functions II:Structural stability: classification and representation

In our previous paper we associated to each non-constant 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 2-dimensional 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

Newton flows for elliptic functions

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

Newton flows for elliptic functions I Structural stability:characterization & genericity

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 (Formula presented.) 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]. They can be described in terms of nondegeneracy-properties of f similar to the rational case [characterization]

A torus bifurcation theorem with symmetry

Hopf bifurcation in the presence of symmetry, in situations where the normal form equations decouple into phase/amplitude equations is described. A theorem showing that in general such degeneracies are expected to lead to secondary torus bifurcations is proved. By applying this theorem to the case of degenerate Hopf bifurcation with triangular symmetry it is proved that in codimension two there exist regions of parameter space where two branches of asymptotically stable two-tori coexist but where no stable periodic solutions are present. Although a theory was not derived for degenerate Hopf bifurcations in the presence of symmetry, examples are presented that would have to be accounted for by any such general theory