Vector fields defined by complex functions

Sverdlove, Ronald

doi:10.1016/0022-0396(79)90029-9

Cited by 22 publications

(28 citation statements)

References 4 publications

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“…More specifically, limit cycles and inverse problems relating to Hilbert's 16th problem which (among other things) considers the upper bound of the number of limit cycles that can occur for real polynomial vector fields in the plane [2,19,20], Hamiltonian systems [19] and the existence of centres [14] are studied. Although no limit cycles exist for holomorphic vector fields, an important strategy when dealing with the Hilbert's 16th problem is to study perturbations of holomorphic vector fields with centres [1,12].…”

Section: History/motivationmentioning

confidence: 99%

Classification of complex polynomial vector fields in one complex variable

Branner

Dias

2010

Journal of Difference Equations and Applications

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Section: History/motivationmentioning

confidence: 99%

Classification of complex polynomial vector fields in one complex variable

Branner

Dias

2010

Journal of Difference Equations and Applications

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“…However, from the point of view of explicit computations, the Darboux centers are hard to find. Instead we will study a particular polynomial holomorphic center of degree n. We recall that an equationż = iz + f (z) has a holomorphic center at the origin when f is a holomorphic function such that f (0) = 0 and Re(f (0)) = 0, see [19]. According to Proposition 3.1 of [6], the holomorphic center is also a Darboux center.…”

Section: Parallelization Of the Lyapunov Constants And Cyclicity For mentioning

confidence: 99%

Parallelization of the Lyapunov constants and cyclicity for centers of planar polynomial vector fields

Liang

Torregrosa

2015

Journal of Differential Equations

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Abstract. Christopher in 2006 proved that under some assumptions the linear parts of the Lyapunov constants with respect to the parameters give the cyclicity of an elementary center. This paper is devote to establish a new approach, namely parallelization, to compute the linear parts of the Lyapunov constants. More concretely, it is showed that parallelization computes these linear parts in a shorter quantity of time than other traditional mechanisms.To show the power of this approach, we study the cyclicity of the holomorphic centerż = iz + z 2 + z 3 + · · · + z n under general polynomial perturbations of degree n, for n ≤ 13. We also exhibit that, from the point of view of computation, among the Hamiltonian, time-reversible, and Darboux centers, the holomorphic center is the best candidate to obtain high cyclicity examples of any degree. For n = 4, 5, . . . , 13, we prove that the cyclicity of the holomorphic center is at least n 2 + n − 2. This result give the highest lower bound for M (6), M (7), . . . , M (13) among the existing results, where M (n) is the maximum number of limit cycles bifurcating from an elementary monodromic singularity of polynomial systems of degree n. As a direct corollary we also obtain the highest lower bound for the Hilbert numbers H(6) ≥ 40, H(8) ≥ 70, and H

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“…The time-p map of (1.1) is used in Sverdlove [1] to establish the nonexistence of limit and semi-limit cycles for (1.1). In particular it is crucially required that the time-p map of (1.1) be holomorphic in Ω.…”

Section: Introductionmentioning

confidence: 99%

“…In particular it is crucially required that the time-p map of (1.1) be holomorphic in Ω. A proof of this statement is not given or referenced in Sverdlove [1], and a proof does not appear to be readily accessible in the relevant literature. The purpose of this note is to provide an elementary proof that the time-p map for (1.1) is holomorphic on Ω.…”

Section: Introductionmentioning

confidence: 99%

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Holomorphic complex differential equations: An elementary proof that the time-p map is holomorphic

Needham

2006

Z. angew. Math. Phys.

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We give an elementary proof that, in its domain of definition, the time-p map of a scalar, autonomous holomorphic, complex differential equation, is itself holomorphic. This result is used by Sverdlove [1] when considering limit cycles in complex holomorphic differential equations. However no proof or reference for the result is given in [1]. Although this result must be well established, a proof does not appear to be readily accessible in the reference literature.

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Vector fields defined by complex functions

Cited by 22 publications

References 4 publications

Classification of complex polynomial vector fields in one complex variable

Classification of complex polynomial vector fields in one complex variable

Parallelization of the Lyapunov constants and cyclicity for centers of planar polynomial vector fields

Holomorphic complex differential equations: An elementary proof that the time-p map is holomorphic

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