具有m条切换线的扰动微分系统的极限环

本文研究具有m条切换线的扰动微分系统的极限环个数问题.所用方法的关键点是通过计算一阶Melnikov函数M(h)的生成元得到M(h)的代数结构.本方法可以用来研究其它具有多条切换线的扰动微分系统的极限环分支问题.     

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.     

Periodic orbits for perturbations of piecewise linear systems

We consider the existence of periodic orbits in a class of three-dimensional piecewise linear systems. Firstly, we describe the dynamical behavior of a non-generic piecewise linear system which has two equilibria and one two-dimensional invariant manifold foliated by periodic orbits. The aim of this work is to study the periodic orbits of the continuum that persist under a piecewise linear perturbation of the system. In order to analyze this situation, we build a real function of real variable whose zeros are related to the limit cycles that remain after the perturbation. By using this function, we state some results of existence and stability of limit cycles in the perturbed system, as well as results of bifurcations of limit cycles. The techniques presented are similar to the Melnikov theory for smooth systems and the method of averaging.     

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).     

Limit cycle bifurcations by perturbing a cuspidal loop in a Hamiltonian system

In this paper, we first study the analytical property of the first Melnikov function for general Hamiltonian systems exhibiting a cuspidal loop and obtain its expansion at the Hamiltonian value corresponding to the loop. Then by using the first coefficients of the expansion we give some conditions for the perturbed system to have 4, 5 or 6 limit cycles in a neighborhood of loop. As an application of our main results, we consider some polynomial Lienard systems and find 4, 5 or 6 limit cycles.     

Limit cycle bifurcations in discontinuous planar systems with multiple lines

In this paper, the limit cycle bifurcation problem is investigated for a class of planar discontinuous perturbed systems with $n$ parallel switch lines. Under the assumption that the unperturbed system has a family of periodic orbits crossing all of the lines, an explicit expression of the first order Melnikov function along the periodic orbits is presented, which plays an important role in studying the problem of limit cycle bifurcations. As an application of the established method, the maximal number of limit cycles of a discontinuous system is considered.     

Limit cycle bifurcation by perturbing a cuspidal loop of order 2 in a Hamiltonian system

This paper deals with the analytical property of the first Melnikov function for general Hamiltonian systems possessing a cuspidal loop of order 2 and its expansion at the Hamiltonian value corresponding to the loop. The explicit formulas for the first coefficients of the expansion have been given. We prove that at least 13 limit cycles can bifurcate from the cuspidal loop of order 2 under certain conditions. Then we consider the cyclicity of a cuspidal loop in some Liénard and Hamiltonian systems, and determine the number of limit cycles that can bifurcate from the perturbed system.     

On the number of limit cycles by perturbing a piecewise smooth Hamilton system with two straight lines of separation

This paper deals with the problem of limit cycle bifurcations for a piecewise smooth Hamilton system with two straight lines of separation. By analyzing the obtained first order Melnikov function, we give upper and lower bounds of the number of limit cycles bifurcating from the period annulus between the origin and the generalized homoclinic loop. It is found that the first order Melnikov function is more complicated than in the case with one straight line of separation and more limit cycles can be bifurcated.     

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.     

Perturbations of May–Leonard system

In this paper we study the quadratic homogeneous perturbations of the 3-dimensional May–Leonard system with α+β=2$\alpha +\beta =2$. It is shown that there are perturbed systems having exactly one or two limit cycles bifurcated from the periodic orbits of May–Leonard system. This is proved by estimating the number of zeros of the first and the second order Melnikov functions.     

Splitting Potential and the Poincaré-Melnikov Method for Whiskered Tori in Hamiltonian Systems

Summary. We deal with a perturbation of a hyperbolic integrable Hamiltonian system with n+1 degrees of freedom. The integrable system is assumed to have n -dimensional hyperbolic invariant tori with coincident whiskers (separatrices). Following Eliasson, we use a geometric approach closely related to the Lagrangian properties of the whiskers, to show that the splitting distance between the perturbed stable and unstable whiskers is the gradient of a periodic scalar function of n phases, which we call splitting potential. This geometric approach works for both the singular (or weakly hyperbolic) case and the regular (or strongly hyperbolic) case, and provides the existence of at least n+1 homoclinic intersections between the perturbed whiskers. In the regular case, we also obtain a first-order approximation for the splitting potential, that we call Melnikov potential. Its gradient, the (vector) Melnikov function, provides a first-order approximation for the splitting distance. Then the nondegenerate critical points of the Melnikov potential give rise to transverse homoclinic intersections between the whiskers. Generically, when the Melnikov potential is a Morse function, there exist at least 2 ⁿ critical points. The first-order approximation relies on the n -dimensional Poincaré-Melnikov method, to which an important part of the paper is devoted. We develop the method in a general setting, giving the Melnikov potential and the Melnikov function in terms of absolutely convergent integrals, which take into account the phase drift along the separatrix and the first-order deformation of the perturbed hyperbolic tori. We provide formulas useful in several cases, and carry out explicit computations that show that the Melnikov potential is a Morse function, in different kinds of examples. Received January 18, 1999; final revision received October 25, 1999; accepted December 12, 1999     

Homoclinic,heteroclinic and periodic orbits of singularly perturbed systems

The main aims of this paper are to study the persistence of homoclinic and heteroclinic orbits of the reduced systems on normally hyperbolic critical manifolds, and also the limit cycle bifurcations either from the homoclinic loop of the reduced systems or from a family of periodic orbits of the layer systems. For the persistence of homoclinic and heteroclinic orbits, and the limit cycles bifurcating from a homolinic loop of the reduced systems, we provide a new and readily detectable method to characterize them compared with the usual Melnikov method when the reduced system forms a generalized rotated vector field. To determine the limit cycles bifurcating from the families of periodic orbits of the layer systems, we apply the averaging methods.We also provide two four-dimensional singularly perturbed differential systems, which have either heteroclinic or homoclinic orbits located on the slow manifolds and also three limit cycles bifurcating from the periodic orbits of the layer system.     

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).     

Limit cycle bifurcations by perturbing piecewise smooth Hamiltonian systems with multiple parameters

This paper is concerned with the problem of limit cycle bifurcation for piecewise smooth near-Hamiltonian systems with multiple parameters. By the first Melnikov function, some novel criteria have been established for the existence of multiple limit cycles. Furthermore, an example is included to validate the obtained results by considering the maximum number of limit cycles for a piecewise quadratic system studied in Llibre and Mereu (2014) [12]. Compared with the result in the above reference, one more limit cycle is found by our method.     

多参数高次扰动的平面近Hamilton系统的极限环个数

本文研究带参数高次扰动的平面近Hamilton系统Melnikov函数,利用一阶Melnikov函数来确定其在Hopf分支中极限环的个数.     

On the Number of Limit Cycles in Small Perturbations of a Piecewise Linear Hamiltonian System with a Heteroclinic Loop

In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n(n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [(n+1)/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in ε. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived.     

Melnikov Functions for a Class of Piecewise Hamiltonian Systems

This paper is concerned with the number of limit cycles for a class of piecewise Hamiltonian systems with two zones separated by two semi-straight lines. By constructing a Poincar\''{e} map, we obtain explicit expressions of the first, second and third order Melnikov functions. In addition, we apply their expressions to give upper bounds of the number of limit cycles bifurcated from a period annulus of a piecewise polynomial Hamiltonian system.     

Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop

In this paper, we study limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with the center at the origin and a homoclinic loop around the origin. By using the first Melnikov function of piecewise near-Hamiltonian systems, we give lower bounds of the maximal number of limit cycles in Hopf and homoclinic bifurcations, and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in ε$\epsilon$. In the case when the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first Melnikov function.     

LIMIT CYCLES OF SOME Z_3-EQUIVARIANT NEAR-HAMILTONIAN SYSTEMS OF DEGREES 3 AND 4

This paper studies the number of limit cycles of some Z3-equivariant near-Hamiltonian systems of degrees 3 and 4,which are a perturbation of a cubic Hamiltonian system. By the Melnikov function method,we obtain 5 and 6 limit cycles respectively.     

Melnikov method for discontinuous planar systems

This paper treats the occurrence of homoclinic solutions in planar systems with discontinuous right-hand side. More precisely, we deal with a T$T$-periodic perturbed system such that the unperturbed system is an autonomous possessing homoclinic orbit. By means of the so-called “non-smooth” Melnikov function there is shown the existence of a homoclinic solution for a perturbed system. The non-smooth Melnikov function is derived, and the method of how to find it in concrete problems is also introduced.