HILBERT'S 16TH PROBLEM AND BIFURCATIONS OF PLANAR POLYNOMIAL VECTOR FIELDS

2003 ◽  
Vol 13 (01) ◽  
pp. 47-106 ◽  
Author(s):  
JIBIN LI
Keyword(s):  
Vector Fields ◽  
Closed Orbit ◽  
Polynomial Vector Fields ◽  
Homoclinic Loop ◽  
Section 8 ◽  
Hilbert’S 16Th Problem ◽  
Hilbert Number ◽  
Fine Focus ◽  
Planar Vector ◽  
Hilbert's 16Th Problem

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.

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.

THE SEPARATRIX VALUES OF A PLANAR HOMOCLINIC LOOP

2009 ◽  
Vol 19 (07) ◽  
pp. 2233-2247 ◽  
Author(s):  
LIQIN ZHAO ◽  
XUEXING WANG
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Homoclinic Bifurcation ◽  
Homoclinic Loop ◽  
Regular Map ◽  
Practical Applications ◽  
The Third ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
The Stability

It is well known that the stability of a homoclinic loop for planar vector fields is closely related to the cyclicity of this homoclinic loop. For a planar homoclinic loop consisting of a hyperbolic saddle, the loop values are crucial to the stability. The loop values are divided into two classes: saddle values and separatrix values. The saddle values are related to Dulac map near the saddle, and the separatrix values are related to the regular map near the homoclinic loop. The alternation of these quantities determines the stability of the homoclinic loop. So, it is important to investigate the separatrix values in both theory and for practical applications. For a given planar vector field, we can try to calculate the saddle values by means of dual Liapunov constants or by finding elementary invariants developed by Liu and Li [1990]. The first separatrix value was obtained by Dulac. The second separatrix value was given by Han and Zhu [2007] and by Hu and Feng [2001] independently. The third separatrix value was obtained by Luo and Li [2005] by means of Tkachev's method. In this paper, we shall establish the formulae for the third and fourth separatrix values. As applications, we will give an example with the homoclinic bifurcation of order 9 and prove that the cyclicity of homoclinic loop together with double homoclinic loops is 57.

Loop Numbers for the Stability of Homoclinic Loops of Planar Vector Fields

2018 ◽  
Vol 28 (08) ◽  
pp. 1850101
Author(s):  
Xingbo Liu ◽  
Xiao-Biao Lin ◽  
Yuzhen Bai ◽  
Deming Zhu
Keyword(s):  
Vector Fields ◽  
Homoclinic Loop ◽  
Nonzero Term ◽  
The Past ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Stability And Bifurcations ◽  
The Stability

This paper is devoted to the study of stability and bifurcations of homoclinic loops for planar vector fields. For a given homoclinic loop, a sequence of loop numbers can be defined such that the stability and bifurcations of the loop are determined by the first nonzero term of the sequence. Formulas for the first several loop numbers were established in the past. In this paper, we will introduce general formulas for the loop numbers for both the single and double homoclinic loops.

INVESTIGATIONS OF BIFURCATIONS OF LIMIT CYCLES IN Z2-EQUIVARIANT PLANAR VECTOR FIELDS OF DEGREE 5

2002 ◽  
Vol 12 (10) ◽  
pp. 2137-2157 ◽  
Author(s):  
JIBIN LI ◽  
H. S. Y. CHAN ◽  
K. W. CHUNG
Keyword(s):  
Dynamical Systems ◽  
Limit Cycles ◽  
Bifurcation Theory ◽  
Vector Fields ◽  
Compound Eyes ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Planar Dynamical Systems

Some distributions of limit cycles of Z2-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 are investigated. These include examples of specific Z2-equivariant fields and Z4-equivariant fields having up to 23 limit cycles. The configurations of compound eyes are also obtained by using the bifurcation theory of planar dynamical systems and the method of detection functions.

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

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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