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.

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.

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.

scholarly journals Integrating Factors and First Integrals for Liénard Type and Frequency-Damped Oscillators

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Emrullah Yaşar
Keyword(s):  
First Integral ◽  
First Integrals ◽  
The Other ◽  
First Order ◽  
Integrating Factor ◽  
Damped Oscillators ◽  
Damped Oscillator ◽  
Integrating Factors ◽  
Oscillator Equations

We consider Liénard type and frequency-damped oscillator equations. Integrating factors and the associated first integrals are derived from the method to compute -symmetries and the associated reduction algorithm. The knowledge of a symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first-order linear systems of partial differential equations. We will compare our results with those gained by the other methods.

scholarly journals Nilpotent centres via inverse integrating factors

2016 ◽  
Vol 27 (5) ◽  
pp. 781-795 ◽  
Author(s):  
ANTONIO ALGABA ◽  
CRISTÓBAL GARCÍA ◽  
JAUME GINÉ
Keyword(s):  
Vector Fields ◽  
Integrating Factor ◽  
The Family ◽  
Formal Integrability ◽  
Formal Inverse ◽  
Inverse Integrating Factor ◽  
Integrating Factors

In this paper, we are interested in the nilpotent centre problem of planar analytic monodromic vector fields. It is known that the formal integrability is not enough to characterize such centres. More general objects are considered as the formal inverse integrating factors. However, the existence of a formal inverse integrating factor is not sufficient to describe all the nilpotent centres. For the family studied in this paper, it is enough.

scholarly journals On the Shape of Limit Cycles That Bifurcate from Isochronous Center

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Guang Chen ◽  
Yuhai Wu
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Asymptotic Expression ◽  
Isochronous Center ◽  
Integrating Factor ◽  
Factor Method ◽  
Inverse Integrating Factor

New idea and algorithm are proposed to compute asymptotic expression of limit cycles bifurcated from the isochronous center. Compared with known inverse integrating factor method, new algorithm to analytically computing shape of limit cycle proposed in this paper is simple and easy to apply. The applications of new algorithm to some examples are also given.

A Class of Three-Dimensional Quadratic Systems with Ten Limit Cycles

2016 ◽  
Vol 26 (09) ◽  
pp. 1650149 ◽  
Author(s):  
Chaoxiong Du ◽  
Yirong Liu ◽  
Wentao Huang
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Three Dimensional ◽  
Singular Value ◽  
Singular Points ◽  
Quadratic Systems ◽  
Symmetric Points ◽  
Fifth Order

Our work is concerned with a class of three-dimensional quadratic systems with two symmetric singular points which can yield ten small limit cycles. The method used is singular value method, we obtain the expressions of the first five focal values of the two singular points that the system has. Both singular symmetric points can be fine foci of fifth order at the same time. Moreover, we obtain that each one bifurcates five small limit cycles under a certain coefficient perturbed condition, consequently, at least ten limit cycles can appear by simultaneous Hopf bifurcation.

The number of limit cycles of certain polynomial differential equations

1984 ◽  
Vol 98 (3-4) ◽  
pp. 215-239 ◽  
Author(s):  
T. R. Blows ◽  
N. G. Lloyd
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Two Dimensional ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Polynomial Differential Equations ◽  
Image Position ◽  
Cubic Systems

SynopsisTwo-dimensional differential systemsare considered, where P and Q are polynomials. The question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of P and Q. An algorithm is described for determining a so-called focal basis; this can be implemented on a computer. Estimates can then be obtained for the number of small-amplitude limit cycles. The technique is applied to certain cubic systems; a class of examples with exactly five small-amplitude limit cycles is constructed. Quadratic systems are also considered.

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