On a Dulac function for the Kukles system

To estimate the number and location of limit cycles of Kukles systems in a strip of the phase plane xOy, we develop a method for constructing the Dulac function in the form of a polynomial in the phase variable y with coefficients depending on the second phase variable x. The suggested method is regular for h ₂(x) = 0. The proof of the fact that the constructed function is a Dulac function is reduced to finding a specific number of positive functions that are linear combinations of known functions of the single phase variable x with arbitrary constants. To construct the Dulac function, we use a solution of the corresponding linear programming problem. In addition, we show that the presented approach is efficient from the practical viewpoint and permits one to obtain a global exact estimate for the number of limit cycles of above-mentioned systems in some cases.     

Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation

This paper intends to explore the bifurcation of limit cycles for planar polynomial systems with even number of degrees. To obtain the maximum number of limit cycles, a sixth-order polynomial perturbation is added to a quintic Hamiltonian system, and both local and global bifurcations are considered. By employing the detection function method for global bifurcations of limit cycles and the normal form theory for local degenerate Hopf bifurcations, 31 and 35 limit cycles and their configurations are obtained for different sets of controlled parameters. It is shown that: H(6)  35 = 6² − 1, where H(6) is the Hilbert number for sixth-degree polynomial systems.     

On the geometry of polynomial dynamical systems

In this paper, the author performs a global qualitative study of plane polynomial dynamical systems and suggests a new geometric approach to solving the sixteenth Hilbert problem on the maximum number and mutual location of their limit cycles in two special cases of such systems. First of all, using the geometric properties of four parameters rotating the vector field of a new canonical system constructed in the paper, the author proposes the proof of his early conjecture, which asserts that the maximum number of limit cycles of an arbitrary quadratic system is equal to 4, and their location (3: 1) is uniquely possible [4]. Then using the same geometric approach, the author solves the primary problem for the polynomial Liénard system (in this special case, it is considered as the thirteenth Smale problem), and generalizing the obtained results, the author formulates the theorem on the maximum number of limit cycles enclosing one singular point in the case of a polynomial system. Moreover, applying the Wintner-Perko termination principle for multiple limit cycles, the author develops an alternative approach to solving the sixteenth Hilbert problem, and using this approach, the author completes the global qualitative study of a general cubic Liénard system having three singular points in the finite part of the plane. In conclusion, the author discusses one more known approach to solving the sixteenth Hilbert problem. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 53, Suzdal Conference-2006, Part 1, 2008.     

Limit cycles for a perturbation of a quadratic center with symmetry

To estimate the number of limit cycles appearing under a perturbation of a quadratic system that has a center with symmetry, we use the method of generalized Dulac functions. To this end, we reduce the perturbed system to a Liénard system with a small parameter, for which we construct a Dulac function depending on the parameter. This permits one to estimate the number of limit cycles in the perturbed system for all sufficiently small parameter values. We find the Dulac function by solving a linear programming problem. The suggested method is used to analyze four specific perturbed systems that globally have exactly three limit cycles [i.e., the limit cycle distribution 3 or (3, 0)] and two systems that have the limit cycle distribution (3, 1) (i.e., one nest around each of the two foci).     

On the number of limit cycles of polynomial Liénard systems

Liénard systems are very important mathematical models describing oscillatory processes arising in applied sciences. In this paper, we study polynomial Liénard systems of arbitrary degree on the plane, and develop a new method to obtain a lower bound of the maximal number of limit cycles. Using the method and basing on some known results for lower degree we obtain new estimations of the number of limit cycles in the systems which greatly improve existing results.     

Limit cycle function of the second kind for autonomous systems on the cylinder

The earlier-developed approach to the solution of the problem of estimating the number of limit cycles and their localization for autonomous systems on the plane with the use of Dulac and Poincaré auxiliary functions is generalized to autonomous systems of differential equations with cylindrical phase surface.     

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     

Dulac-Cherkas function in a neighborhood of a structurally unstable focus of an autonomous polynomial system on the plane

We consider the problem of estimating the number of limit cycles and their localization for an autonomous polynomial system on the plane with fixed real coefficients and with a small parameter. At the origin, the system has a structurally unstable focus whose first Lyapunov focal quantity is nonzero for the zero value of the parameter. We develop an algebraic method for constructing a Dulac-Cherkas function in a neighborhood of this focus in the form of a polynomial of degree 4. The method is based on the construction of an auxiliary positive polynomial containing terms of order ≥ 4 in the phase variables. The coefficients of these terms are found from a linear algebraic system obtained by equating the coefficients of the corresponding auxiliary function with zero. We present examples in which the suggested method permits one to find parameter intervals and the corresponding neighborhoods of the focus in each of which the number of limit cycles remains constant for all parameter values in the respective interval.     

Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation

This paper intends to explore the bifurcation of limit cycles for planar polynomial systems with even number of degrees. To obtain the maximum number of limit cycles, a sixth-order polynomial perturbation is added to a quintic Hamiltonian system, and both local and global bifurcations are considered. By employing the detection function method for global bifurcations of limit cycles and the normal form theory for local degenerate Hopf bifurcations, 31 and 35 limit cycles and their configurations are obtained for different sets of controlled parameters. It is shown that: H(6) ⩾ 35 = 6² − 1, where H(6) is the Hilbert number for sixth-degree polynomial systems.     

On the limit cycles of perturbed discontinuous planar systems with 4 switching lines

Limit cycle bifurcations for a class of perturbed planar piecewise smooth systems with 4 switching lines are investigated. The expressions of the first order Melnikov function are established when the unperturbed system has a compound global center, a compound homoclinic loop, a compound 2-polycycle, a compound 3-polycycle or a compound 4-polycycle, respectively. Using Melnikov’s method, we obtain lower bounds of the maximal number of limit cycles for the above five different cases. Further, we derive upper bounds of the number of limit cycles for the later four different cases. Finally, we give a numerical example to verify the theory results.     

ON THE LIMIT CYCLES OF PLANAR AUTONOMOUS SYSTEMS

91. IntroductionThe theory of ~ cycles is a very aCtive research field of qualitative theory of ordinary~nilal equationS. There have been many mathemsticians studying the nonetistence,~ence and piqueness of ~ cycles for plane systems, and most atteattiope were paidto some special formS (see [2-4, 610] and the references cited therein)' As we know, for thegeneral system on the Planei = p(z,g), b = Q(z,g), (1.1)where P, Q: RZ - R are continuouSly dmerentiable, there are some well-knoWn res…     

Bifurcation of limit cycles near equivariant compound cycles

In this paper we study some equivariant systems on the plane. We first give some criteria for the outer or inner stability of compound cycles of these systems. Then we investigate the number of limit cycles which appear near a compound cycle of a Hamiltonian equivariant system under equivariant perturbations. In the last part of the paper we present an application of our general theory to show that a Z3 equivariant system can have 13 limit cycles.     

Limit cycles under perturbations of a quadratic Hamiltonian system

The perturbed quadratic Hamiltonian system is reduced to a Lienard system with a small parameter for which a Dulac function depending on it is constructed. This permits one to estimate the number of limit cycles of the perturbed system for all sufficiently small parameter values. To find the Dulac function, we use the solution of a linear programming problem. The suggested method is used for studying three specific perturbed systems that have exactly two limit cycles, i.e., the distribution 2 or (0, 2), and one system with distribution (1, 1).     

Normal forms of quadratic vector fields and the Shi Songling equation

We suggest a method that permits one to locate limit cycles of planar vector fields with a weak focus. We use this method to analyze a large domain on the phase plane for the Shi Songling equation. The limit cycles in a neighborhood of the singular point (0, 0) are located in narrow annuli, in which their uniqueness is proved.     

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.     

高阶非线性动力系统全局分析——胞胞映射法应用

本文阐述高阶非线性动力系统全局分析和应用胞胞映射进行分析的一般特点,以及胞胞映射方法对于高阶系统全局分析的有效性;并具体进行了一个弱耦合van der Pol振荡系统的全局分析,确定系统具有两个稳定的极限环,并确定了整个四维空间被分为两个部分,这两部分分别是沿两个极限环运动的渐近稳定域(吸引域).     

Melnikov function and limit cycle bifurcation from a nilpotent center

We investigate a general near-Hamiltonian system on the plane whose unperturbed system has a nilpotent center. Our main purpose is to give an algorithm for calculating the first coefficients of the expansion of the first order Melnikov function. We also give an application by using the method and obtain the number of limit cycles of a cubic system.     

How to estimate the number of limit cycles in Lienard systems with a small parameter

To estimate the number of limit cycles and locate them for polynomial Lienard systems with a small parameter in the case of a perturbation of a center and in the case of the existence of relaxation limit cycles, we develop a method for constructing a modified Dulac function in the form of a series in the small parameter.     

Number and Location of Limit Cycles in a Class of Perturbed Polynomial Systems

In this paper,we investigate the number,location and stability of limit cycles in a class of perturbedpolynomial systems with (2n 1) or (2n 2)-degree by constructing detection function and using qualitativeanalysis.We show that there are at most n limit cycles in the perturbed polynomial system,which is similar tothe result of Perko in [8] by using Melnikov method.For n=2,we establish the general conditions dependingon polynomial's coefficients for the bifurcation,location and stability of limit cycles.The bifurcation parametervalue of limit cycles in [5] is also improved by us.When n=3 the sufficient and necessary conditions for theappearance of 3 limit cycles are given.Two numerical examples for the location and stability of limit cycles areused to demonstrate our theoretical results.     

BIFURCATION THEORY OF QUADRATIC DIFFERENTIAL SYSTEMS

This paper deals with the number of limit cycles and bifurcation problem of quadratic differential systems. Under conditions $a<0,b+2l>0,l+1<0$, the author draws three bifurcation diagrams of the system (1.18) below in the (\delta,m) plane, which show that the maximum number of limit cycles around a focus is two in this case.