Newton flows for elliptic functions I

Helminck, G.F.; Twilt, F.

doi:10.1080/17476933.2017.1350853

Cited by 2 publications

(14 citation statements)

References 16 publications

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“…Moreover, the zeros (poles) for f are just the sinks (sources) of strength 1, whereas the critical points for f are the onefold stagnation points of the stream, compare also [8]. So, the "orthogonal net of the stream-and equipotential-lines" of the planar steady stream is a combination of the phase portraits of N( f ) and N ⊥ ( f ), see Fig.…”

Section: On the Torus T Is Given Bymentioning

confidence: 99%

“…We raised the question whether, and (even so) to what extent, this analogy persists in terms of the corresponding Newton flows (on, respectively, the Riemann sphere S 2 and the torus T ). An affirmative answer to this question is given by comparing the characterization, genericity, classification and representation aspects of rational Newton flows (see [8,Theorem 2.1]) with their counterparts as described in Theorems 1.3, 2.10 and 4.3.…”

Section: Rational Versus Elliptic Newton Flowsmentioning

confidence: 99%

“…3 and 4 planar/toroidal Newton flows for Jacobian functions sn ω 1 ,ω 2 with only simple attractors, repellors and saddles; see also [1,8] and the forthcoming Remark 2.15. For more examples of (structurally stable) Newton flows, see [9].…”

Section: Structural Stabilitymentioning

confidence: 99%

“…We proved (cf. [8] We list some properties that will play a role in the sequel, see the comments on [Duality]…”

Section: Structural Stabilitymentioning

confidence: 99%

“…This representation space can be endowed with a topology, say τ 0 , that is induced by the Euclidean topology on C, and is natural in the following sense (cf. [8]): given an elliptic function f of order r and ε > 0 sufficiently small, there exists a τ 0 -neighbourhood O of f such that for any g ∈ O, the zeros (poles) for g are contained in ε-neighbourhoods of the zeros (poles) for f . …”

Section: The Canonical Form For a Toroidal Newton Flow; The Topology τmentioning

confidence: 99%

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Newton flows for elliptic functions II

Helminck

Twilt

2017

European Journal of Mathematics

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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 the-B Gerard F. Helminck

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Section: On the Torus T Is Given Bymentioning

confidence: 99%

Section: Rational Versus Elliptic Newton Flowsmentioning

confidence: 99%

Section: Structural Stabilitymentioning

confidence: 99%

“…We proved (cf. [8] We list some properties that will play a role in the sequel, see the comments on [Duality]…”

Section: Structural Stabilitymentioning

confidence: 99%

Section: The Canonical Form For a Toroidal Newton Flow; The Topology τmentioning

confidence: 99%

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Newton flows for elliptic functions II

Helminck

Twilt

2017

European Journal of Mathematics

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Newton flows for elliptic functions III & IV

Helminck

Twilt

2018

European Journal of Mathematics

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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 focusses 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.Subject classification: 05C75, 33E05, 34D30, 37C70, 49M15.

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Newton flows for elliptic functions I

Cited by 2 publications

References 16 publications

Newton flows for elliptic functions II

Newton flows for elliptic functions II

Newton flows for elliptic functions III & IV

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