Newton vector fields on the plane and on the torus

Alvarez-Parrilla, Alvaro; Gómez-Arciga, Adrian; Riesgo-Tirado, Alberto

doi:10.1080/17476930902755658

Cited by 4 publications

(14 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Summing up we get the following result (first introduced for (1)(2)(3)(4) in [50], [49], for (5) in the rational case [17], [18] and now expanded to cover (6) …”

Section: Definition 24 1 a Global Analytic Function ξ Is A Collectmentioning

confidence: 99%

“…This technique is used to visualize the examples given in this paper (see [3], [4], [5] for further details).…”

Section: Alvaro Alvarez-parrilla and Jesús Muciño-raymundomentioning

confidence: 99%

“…X is of infinite class if f is of infinite class. 4. For vector fields X of class p, the p-order of X is the corresponding for f , i.e.…”

Section: Example 38 On the Torusmentioning

confidence: 99%

“…The point of view of differential equations (meromorphic vector fields): J. Gregor [26], [27], O. Hájek [31], [32], [33], N. A. Lukashevich [46], L. Brickman et al [14], M. Sabatini [59], J. Muciño-Raymundo et al [50], D. J. Needhan et al [52], E. P. Volokitin et al [70], A. Alvarez-Parrilla et al [4], A. Garijo et al [24], B. Branner et al [13], E. Frías-Armenta et al [23].…”

Section: E(d)mentioning

confidence: 99%

See 3 more Smart Citations

Dynamics of singular complex analytic vector fields with essential singularities I

Alvarez-Parrilla¹,

Muciño–Raymundo²

2017

Conform. Geom. Dyn.

View full text Add to dashboard Cite

Abstract. We tackle the problem of understanding the geometry and dynamics of singular complex analytic vector fields X with essential singularities on a Riemann surface M (compact or not). Two basic techniques are used. (a) In the complex analytic category on M , we exploit the correspondence between singular vector fields X, differential forms ω X (with ω X (X) ≡ 1), orientable quadratic differentials ω X ⊗ ω X , global distinguished parameters Ψ X (z) = z ω X , and the Riemann surfaces R X of the above parameters. (b) We use the fact that all singular complex analytic vector fields can be expressed as the global pullback via certain maps of the holomorphic vector fields on the Riemann sphere, in particular, via their respective Ψ X .We show that under certain analytical conditions on Ψ X , the germ of a singular complex analytic vector field determines a decomposition in angular sectors; center C, hyperbolic H, elliptic E, parabolic P sectors but with the addition of suitable copies of a new type of entire angular sector E , stemming from X(z) = e z ∂ ∂z . This extends the classical theorems of A. A. Andronov et al. on the decomposition in angular sectors of real analytic vector field germs.The Poincaré-Hopf index theory for Re (X) local and global on compact Riemann surfaces, is extended so as to include the case of suitable isolated essential singularities.The inverse problem: determining which cyclic words W X , comprised of hyperbolic, elliptic, parabolic and entire angular sectors, it is possible to obtain from germs of singular analytic vector fields, is also answered in terms of local analytical invariants.We also study the problem of when and how a germ of a singular complex analytic vector field having an essential singularity (not necessarily isolated) can be extended to a suitable compact Riemann surface.Considering the family of entire vector fields E(d) = {X(z) = λe P (z) ∂ ∂z } on the Riemann sphere, where P (z) is a polynomial of degree d and λ ∈ C * , we completely characterize the local and global dynamics of this class of vector fields, compute analytic normal forms for d = 1, 2, 3, and show that for d ≥ 3 there are an infinite number of topological (phase portrait) classes of Re(X), for X ∈ E(d). These results are based on the work of R. Nevanlinna, A. Speisser and M. Taniguchi on entire functions Ψ X .Finally, on the topological decomposition of real vector fields into canonical regions, we extend the results of L. Markus and H. E. Benzinger to meromorphic on C vector fields X, with an essential singularity at ∞ ∈ C, whose Ψ −1 X have d logarithmic branch points over d finite asymptotic values and d logarithmic branch points over ∞.

show abstract

“…Summing up we get the following result (first introduced for (1)(2)(3)(4) in [50], [49], for (5) in the rational case [17], [18] and now expanded to cover (6) …”

Section: Definition 24 1 a Global Analytic Function ξ Is A Collectmentioning

confidence: 99%

“…This technique is used to visualize the examples given in this paper (see [3], [4], [5] for further details).…”

Section: Alvaro Alvarez-parrilla and Jesús Muciño-raymundomentioning

confidence: 99%

“…X is of infinite class if f is of infinite class. 4. For vector fields X of class p, the p-order of X is the corresponding for f , i.e.…”

Section: Example 38 On the Torusmentioning

confidence: 99%

Section: E(d)mentioning

confidence: 99%

See 2 more Smart Citations

Dynamics of singular complex analytic vector fields with essential singularities I

Alvarez-Parrilla¹,

Muciño–Raymundo²

2017

Conform. Geom. Dyn.

View full text Add to dashboard Cite

show abstract

“…In particular, the technique that we present is of interest for people working on vector fields with singularities. For instance, in order to gain insight into the behaviour of analytic vector fields, correct visualization of vector fields in the vicinity of their singular set is required, in the case of visualization of two-dimensional complex analytic vector fields with essential singularities the usual methods only provide partial results see [15][16][17] , whilst the technique which we promote provides accurate and correct solutions 18,19 . These questions arise naturally in discrete and continuous dynamical systems see [20][21][22] .…”

Section: International Journal Of Mathematics and Mathematical Sciencesmentioning

confidence: 99%

On Solving Systems of Autonomous Ordinary Differential Equations by Reduction to a Variable of an Algebra

Alvarez-Parrilla

Frías-Armenta

López‐González

et al. 2012

International Journal of Mathematics and Mathematical Sciences

Self Cite

View full text Add to dashboard Cite

A new technique for solving a certain class of systems of autonomous ordinary differential equations over K n is introduced K being the real or complex field . The technique is based on two observations: 1 , if K n has the structure of certain normed, associative, commutative, and with a unit, algebras A over K, then there is a scheme for reducing the system of differential equations to an autonomous ordinary differential equation on one variable of the algebra; 2 a technique, previously introduced for solving differential equations over C, is shown to work on the class mentioned in the previous paragraph. In particular it is shown that the algebras in question include algebras linearly equivalent to the tensor product of matrix algebras with certain normal forms.

show abstract

El método de algebrización de sistemas de ecuaciones diferenciales

Flores

View full text Add to dashboard Cite

show abstract

Newton vector fields on the plane and on the torus

Cited by 4 publications

References 12 publications

Dynamics of singular complex analytic vector fields with essential singularities I

Dynamics of singular complex analytic vector fields with essential singularities I

On Solving Systems of Autonomous Ordinary Differential Equations by Reduction to a Variable of an Algebra

El método de algebrización de sistemas de ecuaciones diferenciales

Contact Info

Product

Resources

About