Relationships between limit cycles and algebraic invariant curves for quadratic systems

Algebraic limit cycles for quadratic systems started to be studied in 1958. Up to now we know 7 families of quadratic systems having algebraic limit cycles of degree 2, 4, 5 and 6. Here we present some new results on the limit cycles and algebraic limit cycles of quadratic systems. These results provide sometimes necessary conditions and other times sufficient conditions on the cofactor of the invariant algebraic curve for the existence or nonexistence of limit cycles or algebraic limit cycles. In particular, it follows from them that for all known examples of algebraic limit cycles for quadratic systems those cycles are unique limit cycles of the system.     

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.     

THE MAXIMAL NUMBER OF LIMIT CYCLES FOR QUADRATIC DIFFERENTIAL SYSTEM  = -y + lx~2 + xy,  = x(l + ax + by)

In this paper, we investigate the maximal number of limit cycles surrounding a first order weak focus for the quadratic differential system. And proved such a system has at most two limit cycles under some certain conditions.     

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.     

一类平面多项式系统极限环的存在唯一性

研究一类平面2n 1次多项式微分系统的极限环问题,利用Hopf分枝理论得到了该系统极限环存在性与稳定性的若干充分条件,利用Cherkas和Zheilevych的唯一性定理得到了极限环唯一性的若干充分条件.     

一类平面微分系统的极限环

研究一类平面微分系统的极限环,利用Hopf分支理论得到了该系统极限环存在性与稳定性的若干充分条件,利用ЧеркасЛА和ЖилевычЛИ的唯一性定理得到了极限环唯一性的若干充分条件.     

On the number of limit cycles of a Z 4-equivariant quintic near-Hamiltonian system

In this paper, we study the number of limit cycles of a near-Hamiltonian system having Z₄- equivariant quintic perturbations. Using the methods of Hopf and heteroclinic bifurcation theory, we find that the perturbed system can have 28 limit cycles, and its location is also given. The main result can be used to improve the lower bound of the maximal number of limit cycles for some polynomial systems in a previous work, which is the main motivation of the present paper.     

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.     

QUALITATIVE STUDY FOR A CUBIC SYSTEM

This article analyses a non-Lienard type planar cubic system, and a complete qualitative analysis is given for the system, especially the conclusions for the non-existence, existence and uniqueness of limit cycles are obtained.     

On the number of limit cycles for a quintic Li\""{e}nard system under polynomial perturbations

In this paper, we mainly study the number of limit cycles for a quintic Li\"{e}nard system under polynomial perturbations. In some cases, we give new estimations for the lower bound of the maximal number of limit cycles.     

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.     

Limit cycle bifurcations of planar piecewise differential systems with three zones

This paper considers the limit cycle bifurcation problem of planar piecewise differential systems with three zones. Some computation formulas studied the problem of limit cycle bifurcations are provided by introducing multiple parameters. As an application to the obtained method, the number of limit cycles of a piecewise linear system with three zones studied in Lima et al. (2017) is discussed and some more limit cycles are found.     

Limit cycle bifurcations in the in-plane galloping of iced transmission line

In this paper, we establish a mathematical model to describe in-plane galloping of iced transmission line with geometrical and aerodynamical nonlinearities using Hamilton principle. After Galerkin Discretization, we get a two-dimensional ordinary differential equations system, further, a near Hamiltonian system is obtained by rescaling. By calculating the coefficients of the first order Melnikov function or the Abelian integral of the near-Hamiltonian system, the number of limit cycles and their locations are obtained. We demonstrate that this model can have at least 3 limit cycles in some wind velocity. Moreover, some numerical simulations are conducted to verify the theoretical results.     

LIMIT CYCLES AND BIFURCATION CURVES FOR THE QUADRATIC DIFFERENTIAL SYSTEM （Ⅲ）m=0 HAVING THREE ANTI-SADDLES （Ⅱ）

As a continuation of [1], the author studies the limit cycle bifurcation around the focus S1 other than O(0, 0) for the system (1) as δ varies. A conjecture on the non-existence of limit cycles around S1, and another one on the non-coexistence of limit cycles wound both O and S1 are given, together with some numerical examples.     

Effective construction of Poincar\'e--Bendixson regions

This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincar\''e--Bendixson regions by using transversal curves, that enables us to prove the existence of a limit cycle that has been numerically detected. We apply our results to several known systems, like the Brusselator one or some Li\''{e}nard systems, to prove the existence of the limit cycles and to locate them very precisely in the phase space. Our method, combined with some other classical tools can be applied to obtain sharp bounds for the bifurcation values of a saddle-node bifurcation of limit cycles, as we do for the Rychkov system.     

On the cyclicity of a 2-polycycle for quadratic systems

It has been already known that the maximum number of limit cycles near a homoclinic loop of a quadratic Hamiltonian system under quadratic perturbations is two. However, the problem of finding the maximum number of limit cycles in the 2-polycycle case is still open. This paper addresses the problem in some detail and solves it partially.     

UNIQUENESS AND DISTRIBUTION OF LIMIT CYCLES FOR BOUNDED QUADRATIC SYSTEM

1 IntroductionAs we know, any given quadratic system which may have limit cycle (LC,fOr abbreviation) can be written in the fOllowing fOrm (see [1] 512)where 6, l, m, n, a, 6 are all real parameters.If all trajectories of a quadratic system remain bounded fOr t 2 0, we saythat the system is bounded, and fOr abbreviation denote by BQS in this paper.The research work for BQS begin with Dickson-Perko [3]. And then, in [4],they made use of the conclusions of [51 to give a detailed classifica…     

Cubic Lienard Equations with Quadratic Damping (Ⅱ)

Abstract Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienardequations with quadratic damping have at most three limit cycles. This implies that the guess in which thesystem has at most two limit cycles is false. We give the sufficient conditions for the system has at most threelimit cycles or two limit cycles. We present two examples with three limit cycles or two limit cycles by usingnumerical simulation.     

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 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.