scholarly journals A Characterization of Multiplicity-Preserving Global Bifurcations of Complex Polynomial Vector Fields

2020 ◽  
Vol 19 (3) ◽  
Author(s):  
Kealey Dias
Keyword(s):  
Vector Fields ◽  
Complex Polynomial ◽  
Global Bifurcations ◽  
Polynomial Vector ◽  
Polynomial Vector Fields

scholarly journals Finitely curved orbits of complex polynomial vector fields

2007 ◽  
Vol 79 (1) ◽  
pp. 13-16
Author(s):  
Albetã C. Mafra
Keyword(s):  
Vector Field ◽  
Gaussian Curvature ◽  
Algebraic Curve ◽  
Vector Fields ◽  
Complex Polynomial ◽  
Holomorphic Foliations ◽  
Holomorphic Curve ◽  
Polynomial Vector ◽  
Polynomial Vector Fields

This note is about the geometry of holomorphic foliations. Let X be a polynomial vector field with isolated singularities on C². We announce some results regarding two problems: 1. Given a finitely curved orbit L of X, under which conditions is L algebraic? 2. If X has some non-algebraic finitely curved orbit L what is the classification of X? Problem 1 is related to the following question: Let C <FONT FACE=Symbol>Ì</FONT> C² be a holomorphic curve which has finite total Gaussian curvature. IsC contained in an algebraic curve?

scholarly journals Topological enumeration of complex polynomial vector fields

2014 ◽  
Vol 35 (4) ◽  
pp. 1315-1344 ◽  
Author(s):  
J. TOMASINI
Keyword(s):  
Closed Form ◽  
Orthogonal Group ◽  
Vector Fields ◽  
Main Tool ◽  
Roots Of Unity ◽  
Complex Polynomial ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Appl Math ◽  
General Method

AbstractThe enumeration of combinatorial classes of the complex polynomial vector fields in$ \mathbb{C} $presented by K. Dias [Enumerating combinatorial classes of the complex polynomial vector fields in$ \mathbb{C} $.Ergod. Th. & Dynam. Sys. 33(2013), 416–440] is extended here to a closed form enumeration of combinatorial classes for degree$d$polynomial vector fields up to rotations of the$2(d- 1)\mathrm{th} $roots of unity. The main tool in the proof of this result is based on a general method of enumeration developed by V. A. Liskovets [Reductive enumeration under mutually orthogonal group actions.Acta Appl. Math. 52(1998), 91–120].

CENTER PROBLEM AND MULTIPLE HOPF BIFURCATION FOR THE Z5-EQUIVARIANT PLANAR POLYNOMIAL VECTOR FIELDS OF DEGREE 5

2009 ◽  
Vol 19 (06) ◽  
pp. 2115-2121 ◽  
Author(s):  
YIRONG LIU ◽  
JIBIN LI
Keyword(s):  
Vector Field ◽  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Vector Fields ◽  
Center Problem ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Perturbed System ◽  
Lyapunov Constants

This paper proves that a Z5-equivariant planar polynomial vector field of degree 5 has at least five symmetric centers, if and only if it is a Hamltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z5-equivariant perturbations, the conclusion that the perturbed system has at least 25 limit cycles with the scheme 〈5 ∐ 5 ∐ 5 ∐ 5 ∐ 5〉 is rigorously proved.

scholarly journals The Valuative Theory of Foliations

2002 ◽  
Vol 54 (5) ◽  
pp. 897-915 ◽  
Author(s):  
Pedro Fortuny Ayuso
Keyword(s):  
Vector Fields ◽  
Linear Part ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Classical Condition

AbstractThis paper gives a characterization of valuations that follow the singular infinitely near points of plane vector fields, using the notion of L'Hôpital valuation, which generalizes a well known classical condition. With that tool, we give a valuative description of vector fields with infinite solutions, singularities with rational quotient of eigenvalues in its linear part, and polynomial vector fields with transcendental solutions, among other results.

CENTER PROBLEM AND MULTIPLE HOPF BIFURCATION FOR THE Z6-EQUIVARIANT PLANAR POLYNOMIAL VECTOR, FIELDS OF DEGREE 5

2009 ◽  
Vol 19 (05) ◽  
pp. 1741-1749 ◽  
Author(s):  
YIRONG LIU ◽  
JIBIN LI
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Vector Fields ◽  
Hamiltonian Vector Field ◽  
Center Problem ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Perturbed System ◽  
Lyapunov Constants

This paper proves that a Z6-equivariant planar polynomial vector field of degree 5 has at least six symmetric centers, if and only if it is a Hamiltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z6-equivariant perturbations, the conclusion that the perturbed system has at least 24 limit cycles with the scheme 〈 4 ∐ 4 ∐ 4 ∐ 4 ∐ 4 ∐ 4〉 is rigorously proved. Two schemes of distributions of limit cycles are given.

scholarly journals Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three

2021 ◽  
Author(s):  
Niclas Kruff ◽  
Jaume Llibre ◽  
Chara Pantazi ◽  
Sebastian Walcher
Keyword(s):  
Vector Fields ◽  
Arbitrary Dimension ◽  
Classical Problem ◽  
Algebraic Surfaces ◽  
Stationary Points ◽  
Complex Polynomial ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Degree Bounds ◽  
Invariant Algebraic Surfaces

AbstractWe discuss criteria for the nonexistence, existence and computation of invariant algebraic surfaces for three-dimensional complex polynomial vector fields, thus transferring a classical problem of Poincaré from dimension two to dimension three. Such surfaces are zero sets of certain polynomials which we call semi-invariants of the vector fields. The main part of the work deals with finding degree bounds for irreducible semi-invariants of a given polynomial vector field that satisfies certain properties for its stationary points at infinity. As a related topic, we investigate existence criteria and properties for algebraic Jacobi multipliers. Some results are stated and proved for polynomial vector fields in arbitrary dimension and their invariant hypersurfaces. In dimension three we obtain detailed results on possible degree bounds. Moreover by an explicit construction we show for quadratic vector fields that the conditions involving the stationary points at infinity are generic but they do not a priori preclude the existence of invariant algebraic surfaces. In an appendix we prove a result on invariant lines of homogeneous polynomial vector fields.

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