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

Alvarez-Parrilla, Alvaro; Frías-Armenta, Martín Eduardo; López‐González, Elifalet; Yee-Romero, C.

doi:10.1155/2012/753916

Cited by 10 publications

(24 citation statements)

References 22 publications

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“…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: 93%

“…The correspondence of (1) through (5) and the text being: (1) with Theorem 8.16, (2) with Theorem 8.24 and §8. 6.1, (3) and (4) with Theorem 8.31, and (5) with Corollary 10.1.…”

Section: ) a Germ ( C ∞) X Is The Restriction Of X ∈ E(d) If And Omentioning

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%

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Dynamics of singular complex analytic vector fields with essential singularities I

Alvarez-Parrilla¹,

Muciño–Raymundo²

2017

Conform. Geom. Dyn.

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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 ∞.

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“…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: 93%

“…The correspondence of (1) through (5) and the text being: (1) with Theorem 8.16, (2) with Theorem 8.24 and §8. 6.1, (3) and (4) with Theorem 8.31, and (5) with Corollary 10.1.…”

Section: ) a Germ ( C ∞) X Is The Restriction Of X ∈ E(d) If And Omentioning

confidence: 99%

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Dynamics of singular complex analytic vector fields with essential singularities I

Alvarez-Parrilla¹,

Muciño–Raymundo²

2017

Conform. Geom. Dyn.

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“…If = ( , ) has an H(A)-differentiable lifting , we say that planar system (1) is algebrizable and that (2) is an algebrization of (1). A theorem of existence and uniqueness of solutions for differential equations over algebras is proved in [6], and a technique for visualization of solutions is given in [7].…”

Section: Introductionmentioning

confidence: 99%

Algebrization of Nonautonomous Differential Equations

Alcorta-García

Frías-Armenta

Grimaldo-Reyna

et al. 2015

Journal of Applied Mathematics

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Given a planar system of nonautonomous ordinary differential equations, / = ( , ), conditions are given for the existence of an associative commutative unital algebra A with unit and a function : Ω ⊂ R 2 × R 2 → R 2 on an open set Ω such that ( , ) = ( , ) and the maps 1 ( ) = ( , ) and 2 ( ) = ( , ) are Lorch differentiable with respect to A for all ( , ) ∈ Ω, where and represent variables in A. Under these conditions the solutions ( ) of the differential equation / = ( , ) over A define solutions ( ( ), ( )) = ( ) of the planar system.

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“…Si F (f, g) tiene una H (A)−diferenciable sobre H, decimos que el sistema planar (5.0.1) es algebrizable y que (5.0.2) es una algebrización de (5.0.1). Un teorema de existencia y unicidad para ecuaciones diferenciales sobreálgebras es probada en [11], y un método para visualización de soluciones es dada en [4]. En la clásica ecuación diferencial dω/dt = ω 2 tiene soluciones ω(t) = −(t + c) −1 .…”

Section: Algebrización De Ecuaciones Diferenciales No Autónomasunclassified

El método de algebrización de sistemas de ecuaciones diferenciales

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

Cited by 10 publications

References 22 publications

Dynamics of singular complex analytic vector fields with essential singularities I

Dynamics of singular complex analytic vector fields with essential singularities I

Algebrization of Nonautonomous Differential Equations

El método de algebrización de sistemas de ecuaciones diferenciales

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