Newton flows for elliptic functions: a pilot study†

Twilt, F.; Helminck, G.F.; Snuverink, M.; Brug, L. van den

doi:10.1080/02331930701778965

Cited by 4 publications

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References 13 publications

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“…. , r, that fulfil (3). 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.…”

Section: The Canonical Form For a Toroidal Newton Flow; The Topology τ0mentioning

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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“…. , r, that fulfil (3). 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.…”

Section: The Canonical Form For a Toroidal Newton Flow; The Topology τ0mentioning

confidence: 99%

“…1 Tekst rechts onderaan naast plaatje 7 Remark 2.14. For basically the same proof of Corollary 2.13 , see [3].…”

Section: Structurally Stable Elliptic Newton Flows: Classificationunclassified

Newton flows for elliptic functions II

Helminck

Twilt

2017

European Journal of Mathematics

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“…At most one of the partial derivatives ∂ŵ ∂ai (z;ǎ,b), ∂ŵ ∂bj (z;ǎ,b), (z;ǎ,b) ∈ V, i = 1,· · ·, A, j = 1,· · ·, B − 1, vanishes, and thus, in case K > 2: ∂ŵ ∂ai (z;ǎ,b) = 0, ∂ŵ ∂bj (z;ǎ,b) = 0, for at least one i ∈ {1,· · ·, A} or j ∈ {1,· · ·, B − 1.} (21) The latter conclusion cannot be drawn in case K = 2; however, see the forthcoming Remark 5.8. Note that always K 2.…”

Section: Steady Streamsmentioning

confidence: 96%

“…Without loss of generality, we assume (see (21)) that ∂ŵ ∂ai (z 1 ;ǎ,b) = 0. According to the Implicit Function Theorem a local parametrization of Σ around (z 1 ;ǎ,b) exists, given by: (z; a 1 (z, a 2 ,· · ·, a A , b 1 ,· · ·, b B−1 ), a 2 ,· · ·, a A , b 1 ,· · ·, b B−1 ), where a 1 (z 1 , a 2 ,· · ·, a A , b 1 ,· · ·, b B−1 ) = a 1 .…”

Section: Steady Streamsmentioning

confidence: 99%

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Newton flows for elliptic functions I

Helminck

Twilt

2017

Complex Variables and Elliptic Equations

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

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Comment on “Exploring the potential energy landscape of the Thomson problem via Newton homotopies” [J. Chem. Phys. 142, 194113 (2015)]

Bofill

2015

The Journal of Chemical Physics

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Newton flows for elliptic functions: a pilot study†

Cited by 4 publications

References 13 publications

Newton flows for elliptic functions II

Newton flows for elliptic functions II

Newton flows for elliptic functions I

Comment on “Exploring the potential energy landscape of the Thomson problem via Newton homotopies” [J. Chem. Phys. 142, 194113 (2015)]

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