scholarly journals A topological study of planar vector field singularities

2020 ◽  
Vol 40 (9) ◽  
pp. 5217-5245
Author(s):  
Robert Roussarie ◽  
Keyword(s):  
Vector Field ◽  
Planar Vector

ON THE NUMBER AND DISTRIBUTIONS OF LIMIT CYCLES IN A QUINTIC PLANAR VECTOR FIELD

2008 ◽  
Vol 18 (07) ◽  
pp. 1939-1955 ◽  
Author(s):  
YUHAI WU ◽  
YONGXI GAO ◽  
MAOAN HAN
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Hilbert Problem ◽  
Polynomial System ◽  
Polynomial Systems ◽  
Analysis Method ◽  
Quintic Polynomial ◽  
Hilbert Number ◽  
Planar Vector ◽  
16Th Hilbert Problem

This paper is concerned with the number and distributions of limit cycles in a Z2-equivariant quintic planar vector field. By applying qualitative analysis method of differential equation, we find that 28 limit cycles with four different configurations appear in this special planar polynomial system. It is concluded that H(5) ≥ 28 = 52+ 3, where H(5) is the Hilbert number for quintic polynomial systems. The results obtained are useful to the study of the second part of 16th Hilbert problem.

scholarly journals Commuting planar polynomial vector fields for conservative Newton systems

2019 ◽  
Vol 22 (04) ◽  
pp. 1950025 ◽  
Author(s):  
Joel Nagloo ◽  
Alexey Ovchinnikov ◽  
Peter Thompson
Keyword(s):  
Differential Equation ◽  
Vector Field ◽  
Elliptic Equation ◽  
Characteristic Zero ◽  
Classical Result ◽  
Vector Fields ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Linearly Independent ◽  
Planar Vector

We study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field on a plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. Let [Formula: see text], where [Formula: see text] is a field of characteristic zero, and [Formula: see text] the derivation that corresponds to the differential equation [Formula: see text] in a standard way. Let also [Formula: see text] be the Hamiltonian polynomial for [Formula: see text], that is [Formula: see text]. It is known that the set of all polynomial derivations that commute with [Formula: see text] forms a [Formula: see text]-module [Formula: see text]. In this paper, we show that, for every such [Formula: see text], the module [Formula: see text] is of rank [Formula: see text] if and only if [Formula: see text]. For example, the classical elliptic equation [Formula: see text], where [Formula: see text], falls into this category.

Invariant curves and analytic integrability of a planar vector field

2019 ◽  
Vol 266 (2-3) ◽  
pp. 1357-1376 ◽  
Author(s):  
A. Algaba ◽  
C. García ◽  
M. Reyes
Keyword(s):  
Vector Field ◽  
Invariant Curves ◽  
Analytic Integrability ◽  
Planar Vector

Successive derivatives of a first return map, application to the study of quadratic vector fields

1996 ◽  
Vol 16 (1) ◽  
pp. 87-96 ◽  
Author(s):  
J. P. Francoise
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
New Method ◽  
Return Map ◽  
Quadratic Vector Fields ◽  
Planar Vector ◽  
Derivatives Of ◽  
Perturbative Part

AbstractWe provide an algorithm to compute the first non-zero derivative of the return map r ↦ L(r, ε) of a planar vector field which is a polynomial perturbation of . It yields a new method for finding the centre conditions in the case of a homogeneous perturbative part.

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