scholarly journals Center problem for generic degenerate vector fields

2022 ◽  
Vol 214 ◽  
pp. 112597
Author(s):  
Antonio Algaba ◽  
María Díaz ◽  
Cristóbal García ◽  
Jaume Giné
Keyword(s):  
Vector Fields ◽  
Center Problem

scholarly journals A New Normal Form for Monodromic Nilpotent Singularities of Planar Vector Fields

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Antonio Algaba ◽  
Cristóbal García ◽  
Jaume Giné
Keyword(s):  
Normal Form ◽  
Vector Fields ◽  
New Technique ◽  
Center Problem ◽  
New Normal ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
A New Technique

AbstractIn this work, we present a new technique for solving the center problem for nilpotent singularities which consists of determining a new normal form conveniently adapted to study the center problem for this singularity. In fact, it is a pre-normal form with respect to classical Bogdanov–Takens normal formal and it allows to approach the center problem more efficiently. The new normal form is applied to several examples.

scholarly journals Formal Inverse Integrating Factor and the Nilpotent Center Problem

2016 ◽  
Vol 26 (01) ◽  
pp. 1650015 ◽  
Author(s):  
Isaac A. García
Keyword(s):  
Vector Fields ◽  
Center Problem ◽  
Formal Integrability ◽  
Number Formula ◽  
Formal Inverse ◽  
General Object ◽  
Formal Power ◽  
Nilpotent Center ◽  
The Relationship ◽  
Integrating Factors

We are interested in deepening the knowledge of methods based on formal power series applied to the nilpotent center problem of planar local analytic monodromic vector fields [Formula: see text]. As formal integrability is not enough to characterize such a center we use a more general object, namely, formal inverse integrating factors [Formula: see text] of [Formula: see text]. Although by the existence of [Formula: see text] it is not possible to describe all nilpotent centers strata, we simplify, improve and also extend previous results on the relationship between these concepts. We use in the performed analysis the so-called Andreev number [Formula: see text] with [Formula: see text] associated to [Formula: see text] which is invariant under orbital equivalency of [Formula: see text]. Besides the leading terms in the [Formula: see text]-quasihomogeneous expansions that [Formula: see text] can have, we also prove the following: (i) If [Formula: see text] is even and there exists [Formula: see text] then [Formula: see text] has a center; (ii) if [Formula: see text], the existence of [Formula: see text] characterizes all the centers; (iii) if there is a [Formula: see text] with minimum “vanishing multiplicity” at the singularity then, generically, [Formula: see text] has a center.

scholarly journals Orbital Reversibility of Planar Vector Fields

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 14
Author(s):  
Antonio Algaba ◽  
Cristóbal García ◽  
Jaume Giné
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Center Problem ◽  
Normal Form Theory ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Form Theory ◽  
Nilpotent Center

In this work we use the normal form theory to establish an algorithm to determine if a planar vector field is orbitally reversible. In previous works only algorithms to determine the reversibility and conjugate reversibility have been given. The procedure is useful in the center problem because any nondegenerate and nilpotent center is orbitally reversible. Moreover, using this algorithm is possible to find degenerate centers which are orbitally reversible.

scholarly journals The center problem for Z2-symmetric nilpotent vector fields

2018 ◽  
Vol 466 (1) ◽  
pp. 183-198 ◽  
Author(s):  
Antonio Algaba ◽  
Cristóbal García ◽  
Jaume Giné ◽  
Jaume Llibre
Keyword(s):  
Vector Fields ◽  
Center Problem

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.

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.

ON THE CENTER PROBLEM FOR DEGENERATE SINGULAR POINTS OF PLANAR VECTOR FIELDS

2002 ◽  
Vol 12 (04) ◽  
pp. 687-707 ◽  
Author(s):  
VÍCTOR MAÑOSA
Keyword(s):  
Singular Point ◽  
Stability Problem ◽  
Vector Fields ◽  
Blow Up ◽  
Singular Points ◽  
Center Problem ◽  
Focal Value ◽  
The Status ◽  
Planar Vector ◽  
The Stability

The center problem for degenerate singular points of planar systems (the degenerate-center problem) is a poorly-understood problem in the qualitative theory of ordinary differential equations. It may be broken down into two problems: the monodromy problem, to decide if the singular point is of focus-center type, and the stability problem, to decide whether it is a focus or a center. We present an outline on the status of the center problem for degenerate singular points, explaining the main techniques and obstructions arising in the study of the problem. We also present some new results. Our new results are the characterization of a family of vector fields having a degenerate monodromic singular point at the origin, and the computation of the generalized first focal value for this family V1. This gives the solution of the stability problem in the monodromic case, except when V1 = 1. Our approach relies on the use of the blow-up technique and the study of the blow-up geometry of singular points. The knowledge of the blow-up geometry is used to generate a bifurcation of a limit cycle.

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