scholarly journals Non-formally integrable centers admitting an algebraic inverse integrating factor

2018 ◽  
Vol 38 (3) ◽  
pp. 967-988
Author(s):  
Antonio Algaba ◽  
◽  
Natalia Fuentes ◽  
Cristóbal García ◽  
Manuel Reyes
Keyword(s):  
Integrating Factor ◽  
Inverse Integrating Factor

scholarly journals Analytic Integrability of Two Lopsided Systems

2016 ◽  
Vol 26 (02) ◽  
pp. 1650026
Author(s):  
Feng Li ◽  
Pei Yu ◽  
Yirong Liu
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Integrating Factor ◽  
Analytic Integrability ◽  
Inverse Integrating Factor

In this paper, we present two classes of lopsided systems and discuss their analytic integrability. The analytic integrable conditions are obtained by using the method of inverse integrating factor and theory of rotated vector field. For the first class of systems, we show that there are [Formula: see text] small-amplitude limit cycles enclosing the origin of the systems for [Formula: see text], and ten limit cycles for [Formula: see text]. For the second class of systems, we prove that there exist [Formula: see text] small-amplitude limit cycles around the origin of the systems for [Formula: see text], and nine limit cycles for [Formula: see text].

Center cyclicity for some nilpotent singularities including the ℤ2-equivariant class

2020 ◽  
pp. 2050053
Author(s):  
Isaac A. García
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Weighted Degree ◽  
Integrating Factor ◽  
The Family ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Formal Inverse ◽  
Leading Term ◽  
Inverse Integrating Factor

This work concerns with polynomial families of real planar vector fields having a monodromic nilpotent singularity. The families considered are those for which the centers are characterized by the existence of a formal inverse integrating factor vanishing at the singularity with a leading term of minimum [Formula: see text]-quasihomogeneous weighted degree, being [Formula: see text] the Andreev number of the singularity. These families strictly include the case [Formula: see text] and also the [Formula: see text]-equivariant families. In some cases for such families we solve, under additional assumptions, the local Hilbert 16th problem giving global bounds on the maximum number of limit cycles that can bifurcate from the singularity under perturbations within the family. Several examples are given.

scholarly journals Polynomial Inverse Integrating Factors, First Integral and Non-Existence of Limit Cycles in the Plane for Quadratic Systems

2017 ◽  
Vol 5 (2) ◽  
pp. 232
Author(s):  
Ahmed M. Hussien
Keyword(s):  
Limit Cycles ◽  
First Integral ◽  
Quadratic Systems ◽  
Integrating Factor ◽  
Inverse Integrating Factor ◽  
Integrating Factors

The main purpose of this paper is to study the existence of polynomial inverse integrating factor and first integral, and non-existence of limit cycles for all systems. Furthermore, we consider some applications.

PHASE PORTRAITS OF THE QUADRATIC SYSTEMS WITH A POLYNOMIAL INVERSE INTEGRATING FACTOR

2009 ◽  
Vol 19 (03) ◽  
pp. 765-783 ◽  
Author(s):  
BARTOMEU COLL ◽  
ANTONI FERRAGUT ◽  
JAUME LLIBRE
Keyword(s):  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Polynomial Differential Systems ◽  
Integrating Factor ◽  
Inverse Integrating Factor

We classify the phase portraits of all planar quadratic polynomial differential systems having a polynomial inverse integrating factor.

ON THE CENTERS OF PLANAR ANALYTIC DIFFERENTIAL SYSTEMS

2007 ◽  
Vol 17 (09) ◽  
pp. 3061-3070 ◽  
Author(s):  
JAUME GINÉ
Keyword(s):  
Point Of View ◽  
Factor V ◽  
Theoretical Point ◽  
Linear Type ◽  
Differential Systems ◽  
Integrating Factor ◽  
Inverse Integrating Factor

In this work we continue the study of the centers which are limits of linear type centers. It is proved that if a degenerate center has an inverse integrating factor V(x, y) with V(0, 0) ≠ 0, then this degenerate center is also the limit of linear type centers. Moreover, we show that the degenerate centers with characteristic directions that are the limits of degenerate centers without characteristic directions are also detectable, from a theoretical point of view, with the Bautin method.

scholarly journals A Survey on the Inverse Integrating Factor

2010 ◽  
Vol 9 (1-2) ◽  
pp. 115-166 ◽  
Author(s):  
Isaac A. García ◽  
Maite Grau
Keyword(s):  
Integrating Factor ◽  
Inverse Integrating Factor

scholarly journals A class of non-integrable systems admitting an inverse integrating factor

2014 ◽  
Vol 420 (2) ◽  
pp. 1439-1454 ◽  
Author(s):  
A. Algaba ◽  
N. Fuentes ◽  
C. García ◽  
M. Reyes
Keyword(s):  
Integrable Systems ◽  
Integrating Factor ◽  
Inverse Integrating Factor
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