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).     

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.     

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.     

Limit cycles and invariant cylinders for a class of continuous and discontinuous vector field in dimention 2n

The subject of this paper concerns with the bifurcation of limit cycles and invariant cylinders from a global center of a linear differential system in dimension 2n perturbed inside a class of continuous and discontinuous piecewise linear differential systems. Our main results show that at most one limit cycle and at most one invariant cylinder can bifurcate using the expansion of the displacement function up to first order with respect to a small parameter. This upper bound is reached. For proving these results we use the averaging theory in a form where the differentiability of the system is not needed.     

Dulac–Cherkas criterion for exact estimation of the number of limit cycles of autonomous systems on a plane

The problem of exact nonlocal estimation of the number of limit cycles surrounding one point of rest in a simply connected domain of the real phase space is considered for autonomous systems of differential equations with continuously differentiable right-hand sides. Three approaches to solving this problem are proposed that are based on sequential two-step usage of the Dulac–Cherkas criterion, which makes it possible to find closed transversal curves dividing the connected domain in doubly connected subdomains that surround the point of rest, with the system having precisely one limit cycle in each of them. The effectiveness of these approaches is exemplified with polynomial Liènard systems, a generalized van der Pol system, and a perturbed Hamiltonian system. For some systems, the derived estimate holds true in the entire phase space.     

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.     

Bendixson-Dulac criterion and reduction to global uniqueness in the problem of estimating the number of limit cycles

For autonomous systems on the real plane, we develop a regular method for localizing and estimating the number of limit cycles surrounding the unique singular point. The method is to divide the phase plane into annulus-shaped domains with transversal boundaries in each of which a Dulac function is constructed by solving an optimization problem, which permits one to use the Bendixson-Dulac criterion. We state the principle of reduction to global uniqueness and use it in the case of existence of an Andronov-Hopf function of limit cycles to obtain a sharp global estimate of the number of limit cycles for an individual system as well as for a one-parameter family of such systems in an unbounded domain.     

One‐parametric families of canard cycles: two explicitly solvable examples

Systems of singularly perturbed autonomous ordinary differential equations possessing in a parameter plane two intersecting bifurcation curves connected with the generation of limit cycles with large and small amplitude respectively, have a special class of limit cycles called canards or french ducks describing an exponentially fast transition from a small amplitude limit cycle to limit cycle with a large amplitude. We present two explicitly integrable examples of non‐autonomous singularly perturbed di.erential equations with canard cycles without a second parameter.     

The Same Distribution of Limit Cycles in a Hamiltonian System with Nine Seven-order Perturbed Terms

Using qualitative analysis and numerical simulation, we investigate the number and distribution of limit cycles for a cubic Hamiltonian system with nine different seven-order perturbed terms. It is showed that these perturbed systems have the same distribution of limit cycles. Furthermore, these systems have 13, 11 and 9 limit cycles for some parameters, respectively. The accurate positions of the 13, 11 and 9 limit cycles are obtained by numerical exploration, respectively. Our results imply that these perturbed systems are equivalent in the sense of distribution of limit cycles. This is useful for studying limit cycles of perturbed systems.     

一类二维系统的极限环的个数

本文利用Dulac函数方法讨论一类二维系统在环城上的包围多个奇点的极限环的唯一性及在n连城上极限环的唯n-1性，并给出了两个多项式的例子，讨论了极限环的唯一性和唯二性．     

Bifurcation of limit cycles from a 4-dimensional center in 1:n resonance

We study the bifurcation of limit cycles from the periodic orbits of a linear differential system in R⁴ in resonance 1:n perturbed inside a class of piecewise linear differential systems, which appear in a natural way in control theory. Our main result shows that at most 1 limit cycle can bifurcate using expansion of the displacement function up to first order with respect to a small parameter. This upper bound is reached. For proving this result we use the averaging theory in a form where the differentiability of the system is not needed.     

一类极限环唯一性的充分条件

包围多个奇点的极限环的唯一性给出一组简洁的充分条件,并将它应用于几类非线性振动方程及多项式微分系统.最后通过系统(1)我们指出证明极限环唯一性中的几种常用方法之间的内在联系,并指出对形如(1)的系统,作 Dulac 函数的一般规律.假设(1)中函数对一切变元连续且满足初值解的唯一性条件.若涉及它们的导数或     

Investigation of relation between singular points and number of limit cycles for a rotor–AMBs system

The relation between singular points and the number of limit cycles is investigated for a rotor-active magnetic bearings system with time-varying stiffness and single-degree-of-freedom. The averaged equation of the system is a perturbed polynomial Hamiltonian system of degree 5. The dynamic characteristics of the unperturbed system are first analyzed for a certain parameter group. The number of limit cycles and their configurations of the perturbed system under eight different parametric groups are obtained and the influence of eight control conditions on the number of limit cycles is studied. The results obtained here will play an important leading role in the study of the properties of nonlinear dynamics and control of the rotor-active magnetic bearings system with time-varying stiffness.     

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.     

MELNIKOV FUNCTIONS AND PERTURBATION OF A PLANAR HAMILTONIAN SYSTEM

In this paper, Melnikov functions which apper in the study of limit cycles of a perturbed planar Hamiltonlan system are studied. There are two main contributions here. The first one is related to the explicit formulae for these functions: a new method is developed to achieve the goal for the second one (Theorem A). the authors also discover a close relation between Melnlkov functions and focal qtmntities (Theorem 13). This relation is useful in both judging when a Melnikov function is identically zero and simplifying the computation of a Melnikov function (see §5). I)espite these results, this paper also includes other related resuEs, e.g. some estimations of the upper bound for the number of limit cycles in a perturbed Hamiltonian system.     

SAME DISTRIBUTION OF LIMIT CYCLES TO PERTURBED CUBIC HAMILTONIAN SYSTEMS WITH DIFFERENT PERTURBATIONS

Using qualitative analysis, we study perturbed Hamiltonian systems with different n-th order polynomial as perturbation terms. By numerical simulation, we show that these perturbed systems have the same distribution of limit cycles. Our results imply that these perturbed systems are equivalent in the sense of distribution of limit cycles. This is useful for studying limit cycles of perturbed systems.     

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.     

THE PROBLEM OF LIMIT CYCLES FOR A LIENARD TYPE CUBIC SYSTEM

The purpose of this paper is to study a general Lienard type cubic system with one antisaddle and two saddles. We give some results of the existence and uniqueness of limit cycles as well as the evolution of limit cycles around the antisaddle for system (2) in the following when parameter a1 changes.     

Dulac的构造及其应用(Ⅱ)

本文研究（Ⅱ）类系统Dulac函数的构造问题,作为应用,讨论此系统的极限环的不存在性问题,得到了一些新结果.