A Restricted Version of Hilbert's 16th Problem for Quadratic Vector Fields

The original Hilbert's 16th problem can be split into four parts consisting of Problems A–D. In this paper, the progress of study on Hilbert's 16th problem is presented, and the relationship between Hilbert's 16th problem and bifurcations of planar vector fields is discussed. The material is presented in eight sections. Section 1: Introduction: what is Hilbert's 16th problem? Section 2: The first part of Hilbert's 16th problem. Section 3: The second part of Hilbert's 16th problem: introduction. Section 4: Focal values, saddle values and finite cyclicity in a fine focus, closed orbit and homoclinic loop. Section 5: Finiteness problem. Section 6: The weakened Hilbert's 16th problem. Section 7: Global and local bifurcations of Zq–equivariant vector fields. Section 8: The rate of growth of Hilbert number H(n) with n.

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Hilbert’s 16th problem on a period annulus and Nash space of arcs

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004119000239 ◽

2019 ◽

Vol 169 (2) ◽

pp. 377-409

Author(s):

JEAN–PIERRE FRANÇOISE ◽

LUBOMIR GAVRILOV ◽

DONGMEI XIAO

Keyword(s):

Normal Form ◽

Vector Fields ◽

Exceptional Divisor ◽

Blow Up ◽

Hilbert’S 16Th Problem ◽

Period Annulus ◽

Irreducible Components ◽

Geometric Setting ◽

Bautin Ideal ◽

Hilbert's 16Th Problem

AbstractThis paper introduces an algebro-geometric setting for the space of bifurcation functions involved in the local Hilbert’s 16th problem on a period annulus. Each possible bifurcation function is in one-to-one correspondence with a point in the exceptional divisor E of the canonical blow-up BI ℂn of the Bautin ideal I. In this setting, the notion of essential perturbation, first proposed by Iliev, is defined via irreducible components of the Nash space of arcs Arc(BI ℂn, E). The example of planar quadratic vector fields in the Kapteyn normal form is further discussed.

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On the singularity structure of Kahan discretizations of a class of quadratic vector fields

European Journal of Mathematics ◽

10.1007/s40879-021-00479-4 ◽

2021 ◽

Author(s):

René Zander

Keyword(s):

Elliptic Curves ◽

Vector Fields ◽

Singularity Structure ◽

Quadratic Vector Fields ◽

Parameter Values

AbstractWe discuss the singularity structure of Kahan discretizations of a class of quadratic vector fields and provide a classification of the parameter values such that the corresponding Kahan map is integrable, in particular, admits an invariant pencil of elliptic curves.

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Planar quadratic vector fields with finite saddle connection on a straight line (convex case)

Qualitative Theory of Dynamical Systems ◽

10.1007/bf02972672 ◽

2005 ◽

Vol 6 (2) ◽

pp. 187-204 ◽

Cited By ~ 1

Author(s):

Paulo César Carrião ◽

Maria Elasir Seabra Gomes ◽

Antonio Augusto Gaspar Ruas

Keyword(s):

Vector Fields ◽

Saddle Connection ◽

Quadratic Vector Fields ◽

Straight Line ◽

Convex Case

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Twelve Limit Cycles in 3D Quadratic Vector Fields with Z3 Symmetry

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418501390 ◽

2018 ◽

Vol 28 (11) ◽

pp. 1850139 ◽

Cited By ~ 3

Author(s):

Laigang Guo ◽

Pei Yu ◽

Yufu Chen

Keyword(s):

Limit Cycles ◽

Vector Fields ◽

Center Manifold ◽

Three Dimensional ◽

Center Manifold Theory ◽

Normal Form Theory ◽

Quadratic Vector Fields ◽

Fine Focus ◽

Form Theory ◽

Manifold Theory

This paper is concerned with the number of limit cycles bifurcating in three-dimensional quadratic vector fields with [Formula: see text] symmetry. The system under consideration has three fine focus points which are symmetric about the [Formula: see text]-axis. Center manifold theory and normal form theory are applied to prove the existence of 12 limit cycles with [Formula: see text]–[Formula: see text]–[Formula: see text] distribution in the neighborhood of three singular points. This is a new lower bound on the number of limit cycles in three-dimensional quadratic systems.

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