在混合扰动下从闭轨族分支的极限环

本文讨论了对确定微分方程组的向量场和分析其轨线穿过方向的参考闭曲线族同时进行扰动分支极限环的方法，并给出了一个平面二次微分系统在混合扰动下分支出三个极限环的例子     

POLYNOMIAL INVERSE INTEGRATING FACTORS

1IntroductionWeconsiderinthispapertwo-dimensionalautonomoussystemsofdifferentialequationsoftheformwhere'=2,p(x,y)andQ(x,y)arepolynomials,i.e.wherePh(x,y)andoh(x,y)arehomogeneouspolynomialsofdegreek.Thetwofundamentalproblemsofthequalitativetheoryofsystem(1)aretheproblemofthecenterandthedeterminationofthenumberoflimitcyclesandtheirlocationinphasespace.Inrecentworksithasbeenshownthataunifiedmethodcanbeusedtostudytheseproblems[2],[3],[4],[6],[8]and[12](andreferencestherein).Themethodisbasedonthed…     

Bifurcation and Isochronicity at Infinity in a Class of Cubic Polynomial Vector Fields

In this paper, we study the appearance of limit cycles from the equator and isochronicity of infinity in polynomial vector fields with no singular points at infinity. We give a recursive formula to compute the singular point quantities of a class of cubic polynomial systems, which is used to calculate the first seven singular point quantities. Further, we prove that such a cubic vector field can have maximal seven limit cycles in the neighborhood of infinity. We actually and construct a system that has seven limit cycles. The positions of these limit cycles can be given exactly without constructing the Poincare cycle fields. The technique employed in this work is essentially different from the previously widely used ones. Finally, the isochronous center conditions at infinity are given.     

The loop quantities and bifurcations of homoclinic loops

The stability and bifurcations of a homoclinic loop for planar vector fields are closely related to the limit cycles. For a homoclinic loop of a given planar vector field, a sequence of quantities, the homoclinic loop quantities were defined to study the stability and bifurcations of the loop. Among the sequence of the loop quantities, the first nonzero one determines the stability of the homoclinic loop. There are formulas for the first three and the fifth loop quantities. In this paper we will establish the formula for the fourth loop quantity for both the single and double homoclinic loops. As applications, we present examples of planar polynomial vector fields which can have five or twelve limit cycles respectively in the case of a single or double homoclinic loop by using the method of stability-switching.     

THE NUMBER OF LIMIT CYCLESOF THE PLANE AUTONOMOUS SYSTEMS

In this paper the author discusses the vector fields satisfying the conditions (C1) and (C2) and proves some theorems for the number of limit cycles of the plane autonomous systems. Some applications of the theorems and lemmas in §1 are given in §2.     

Quadratic systems with maximum number of limit cycles

We suggest a method for obtaining quadratic systems with a given distribution of limit cycles. We use it to obtain a set of quadratic systems with the distributions (3, 1), (3, 0), and 3 of limit cycles and with different configurations of singular points. The distributions are justified with the use of a modified Dulac function in a natural domain of existence of limit cycles.     

Abelian integrals and bifurcation theory

Conditions are given for uniqueness of limit cycles for autonomous equations in the plane. The results are applicable to codimension two bifurcations near equilibrium points for vector fields.     

Perturbation from an elliptic Hamiltonian of degree four—III global centre

The paper deals with Liénard equations of the form , with P and Q polynomials of degree, respectively, 3 and 2. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree four, exhibiting a global centre. It is proven that the least upper bound of the number of zeros of the related elliptic integral is four, and this is a sharp one.This result permits to prove the existence of Liénard equations of type (3,2) with a quadruple limit cycle, with both a triple and a simple limit cycle, with two semistable limit cycles, with one semistable and two simple limit cycles or with four simple limit cycles.     

On the number of limit cycles of some systems on the cylinder

In this article we give two criteria for bounding the number of non-contractible limit cycles of a family of differential systems on the cylinder. This family includes Abel equations as well as the polar expression of several types of planar polynomial systems given by the sum of three homogeneous vector fields.     

Perturbation from an asymmetric cubic Hamiltonian

This paper concerns with the number of limit cycles from an asymmetric Hamiltonian of degree three under cubic perturbation. Eleven limit cycles are found and three different distributions are given by using the methods of bifurcation theory and qualitative analysis, two of which are new.     

On the Bifurcations of a Hamiltonian Having Three Homoclinic Loops under Z3 Invariant Quintic Perturbations

A cubic system having three homoclinic loops perturbed by Z3 invariant quintic polynomials is considered. By applying the qualitative method of differential equations and the numeric computing method, the Hopf bifurcation, homoclinic loop bifurcation and heteroclinic loop bifurcation of the above perturbed system are studied. It is found that the above system has at least 12 limit cycles and the distributions of limit cycles are also given.     

Small limit cycles bifurcating from fine focus points in quartic order Z3-equivariant vector fields

This paper is concerned with the number of limit cycles for a quartic polynomial Z₃-equivariant vector fields. The system under consideration has a fine focus point at the origin, and three fine focus points which are symmetric about the origin. By the computation of the singular point values, sixteen limit cycles are found and their distributions are studied by using the new methods of bifurcation theory and qualitative analysis. This is a new result in the study of the second part of the 16th Hilbert problem. It gives rise to the conclusion: H(4)?16, where H(n) is the Hilbert number for the second part of Hilbert's 16th problem. The process of the proof is algebraic and symbolic. As far as know, the technique employed in this work is different from more usual ones, the calculation can be readily done with using computer symbol operation system such as Mathematica.     

Bifurcations of limit cycles in a Z6-equivariant planar vector field of degree 5

A concrete numerical example of Z6-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 having at least 24 limit cycles and the configurations of compound eyes are given by using the bifurcation theory of planar dynamical systems and the method of detection functions. There is reason to conjecture that the Hilbert number H(2k + 1) ≥ (2k + 1)2 - 1 for the perturbed Hamiltonian systems.     

The 16th Hilbert problem on algebraic limit cycles

For real planar polynomial differential systems there appeared a simple version of the 16th Hilbert problem on algebraic limit cycles: Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m? In [J. Llibre, R. Ramírez, N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations 248 (2010) 1401-1409] Llibre, Ramírez and Sadovskaia solved the problem, providing an exact upper bound, in the case of invariant algebraic curves generic for the vector fields, and they posed the following conjecture: Is1+(m−1)(m−2)/2the maximal number of algebraic limit cycles that a polynomial vector field of degree m can have?In this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves. This result includes the Llibre et al.?s as a special one. For the polynomial vector fields having only non-dicritical invariant algebraic curves we answer the simple version of the 16th Hilbert problem.     

The number and distributions of limit cycles for a class of cubic near-Hamiltonian systems

This paper concerns with the number of limit cycles for a cubic Hamiltonian system under cubic perturbation. The fact that there exist 9-11 limit cycles is proved. The different distributions of limit cycles are given by using methods of bifurcation theory and qualitative analysis, among which two distributions of eleven limit cycles are new.     

The bifurcation of limit cycles in Zn-equivariant vector fields

In this paper, we study the bifurcation of limit cycles from fine focus in Z_n-equivariant vector fields. An approach for investigating bifurcation was obtained. In order to show our work is efficacious, an example on bifurcations behavior is given, namely five order singular points values are given in the seventh degree Z₈-equivariant systems. We discuss their bifurcation behavior of limit cycles, and show that there are eight fine focuses of five order and five small amplitude limit cycles can bifurcate from each. So 40 small amplitude limit cycles can bifurcate from eight fine focuses under a certain condition. In terms of the number of limit cycles for seventh degree Z₈-equivariant systems, our results are good and interesting.     

Invariant circles for homogeneous polynomial vector fields on the 2-dimensional sphere

LexX be a homogeneous polynomial vector field of degreen≥3 on S² having finitely many invariant circles. Then, for such a vector fieldX we find upper bounds for the number of invariant circles, invariant great circles, invariant circles intersecting at a same point and parallel circles with the same director vector. We give examples of homogeneous polynomial vector fields of degree 3 on S² having finitely many invariant circles which are not great circles, which are limit cycles, but are not great circles and invariant great circles that are limit cycles. Moreover, for the casen=3 we determine the maximum number of parallel invariant circles with the same director vector. The authors are partially supported by a MCYT grant BFM2002-04236-C02-02 and by a CIRIT grant number 2001SGR 00173.     

On the limit cycles of a quintic planar vector field

This paper concerns the number and distributions of limit cycles in a Z_2-equivariant quintic planar vector field.25 limit cycles are found in this special planar polynomial system and four different configurations of these limit cycles are also given by using the methods of the bifurcation theory and the qualitative analysis of the differential equation.It can be concluded that H(5)≥25=5~2, where H(5)is the Hilbert number for quintic polynomial systems.The results obtained are useful to study the weakened 16th Hilbert problem.     

Ten limit cycles around a center-type singular point in a 3-d quadratic system with quadratic perturbation

In this paper, we show that perturbing a simple 3-d quadratic system with a center-type singular point can yield at least 10 small-amplitude limit cycles around a singular point. This result improves the 7 limit cycles obtained recently in a simple 3-d quadratic system around a Hopf singular point. Compared with Bautin’s result for quadratic planar vector fields, which can only have 3 small-amplitude limit cycles around an elementary center or focus, this result of 10 limit cycles is surprisingly high. The theory and methodology developed in this paper can be used to consider bifurcation of limit cycles in higher-dimensional systems.     

Bifurcations of limit cycles in a Z6-equivariant planar vector field of degree 5

A concrete numerical example of Z₆-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 having at least 24 limit cycles and the configurations of compound eyes are given by using the bifurcation theory of planar dynamical systems and the method of detection functions. There is reason to conjecture that the Hilbert number H(2k + 1) ⩾ (2k + I)² - 1 for the perturbed Hamiltonian systems.