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…     

Proof of Artés–Llibre–Valls's conjectures for the Higgins–Selkov and the Selkov systems

The aim of this paper is to prove Artés–Llibre–Valls's conjectures on the uniqueness of limit cycles for the Higgins–Selkov system and the Selkov system. In order to apply the limit cycle theory for Liénard systems, we change the Higgins–Selkov and the Selkov systems into Liénard systems first. Then, we present two theorems on the nonexistence of limit cycles of Liénard systems. At last, the conjectures can be proven by these theorems and some techniques applied for Liénard systems.     

Liénard方程极限环的存在性

本文在没有常设条件G(±∞)=+∞的情况下,证明了Liénard方程存在极限环的几个充分性定理,推广了文[3～6]的某些结果.这些定理给出的条件均可估计极限环的存在区域.至少在n个极限环的充分性定理3、4的条件既不要求F(x)是奇函数,也不要求F(x)"n重互相相容"或"n重互相包含".     

The limit cycles of a second-order system of differential equations: The method of small forms

In this paper we investigate the existence of limit cycles of a system of the second-order differential equations with a vector parameter.We propose a method for representing a solution as a sum of forms with respect to the initial value and the parameter; we call this technique the method of small forms. We establish the conditions under which a sufficiently small neighborhood of the equilibrium point contains no limit cycles. We construct a polynomial, whose positive roots of odd multiplicity define the lower bound for the number of cycles, and simple positive roots (other positive roots do not exist) define the number of limit cycles in a sufficiently small neighborhood of the equilibrium point.We prove theorems, whose conditions guarantee that a positive root of odd multiplicity defines a unique limit cycle, but a positive root of even multiplicity defines exactly two limit cycles.We propose a method for defining the type of the stability of limit cycles.     

Oscillations in some nonlinear economic relationships

In this paper we consider a simple family of nonlinear dynamical systems generated by smooth functions. Some theorems for the existence and the uniqueness of the limit cycles of the systems are presented. If f and g are generating functions with unique limit cycles and xf(x) < xg(x), for all x ≠ 0, then according to the ‘bounding theorem’ proved in the paper, the limit cycle of the system generated by f is bounded by the limit cycle of the system generated by g. This gives us a method to estimate the amplitude of the oscillations also for systems for which we do not know the generating function exactly. As an application we extend the nonlinear business cycle model proposed by Tönu Puu (1989).     

Limit cycle bifurcations of some Liénard systems

In this paper we first give some general theorems on the limit cycle bifurcation for near-Hamiltonian systems near a double homoclinic loop or a center as a preliminary. Then we use these theorems to study some polynomial Liénard systems with perturbations and give new lower bounds for the maximal number of limit cycles of these systems.     

一类脉冲微分系统的有界变差解

在比文[6]更弱的条件下讨论了固定时刻脉冲微分系统与Kurzweil广义常微分方程的关系，并建立了这类脉冲微分系统有界变差解的局部存在性和唯一性定理．     

The planar discontinuous piecewise linear refracting systems have at most one limit cycle

In this paper we investigate the limit cycles of planar piecewise linear differential systems with two zones separated by a straight line. It is well known that when these systems are continuous they can exhibit at most one limit cycle, while when they are discontinuous the question about maximum number of limit cycles that they can exhibit is still open. For these last systems there are examples exhibiting three limit cycles.The aim of this paper is to study the number of limit cycles for a special kind of planar discontinuous piecewise linear differential systems with two zones separated by a straight line which are known as refracting systems. First we obtain the existence and uniqueness of limit cycles for refracting systems of focus-node type. Second we prove that refracting systems of focus–focus type have at most one limit cycle, thus we give a positive answer to a conjecture on the uniqueness of limit cycle stated by Freire, Ponce and Torres in Freire et al. (2013). These two results complete the proof that any refracting system has at most one limit cycle.     

Order-Preserving Skew-Product Flows and Nonautonomous Parabolic Systems

The paper deals with order-preserving (or monotone) skew-product flows in Banach spaces. Within a general framework, we study long-time behavior of their trajectories. We introduce the concepts of equilibria, sub- and super-equilibria and we give simple conditions that guarantee their existence. Attention is mainly paid to order-preserving skew-product flows which have an additional concavity property called sublinearity, frequently encountered in applications. For these flows we prove the uniqueness and stability of equilibria and also a limit set trichotomy, stating that either (i) all orbits are unbounded, or (ii) all orbits are bounded but their closure reaches out to the boundary of the part, or (iii) there exists a unique, globally attracting equilibrium. Our main examples are quasilinear systems of parabolic equations with almost-periodic coefficients.     

On a family of models of cell division cycle

The aim of this work is to generalize and study a model of cell division cycle proposed recently by Zheng et al. [Zheng Z, Zhou T, Zhang S. Dynamical behavior in the modeling of cell division cycle. Chaos, Solitons & Fractals 2000;11:2371–8]. Here we study the qualitative properties of a general family to which the above model belongs. The global asymptotic stability (GAS) of the unique equilibrium point E (idest of the arrest of cell cycling) is investigated and some conditions are given. Hopf’s bifurcation is showed to happen. In the second part of the work, the theorems given in the first part are used to analyze the GAS of E and improved conditions are given. Theorem on uniqueness of limit cycle in Lienard’s systems are used to show that, for some combination of parameters, the model has GAS limit cycles.     

The improved LaSalle-type theorems for stochastic functional differential equations

The main aim of this paper is to improve some results obtained by Mao [X. Mao, The LaSalle-type theorems for stochastic functional differential equations, Nonlinear Stud. 7 (2000) 307-328]. Our new theorems give better results while conditions imposed are much weaker than in the paper mentioned above. For example, we need only the local Lipschitz condition but neither the linear growth condition nor the bounded moment condition on the solutions. To guarantee the existence and uniqueness of the global solution to the underlying stochastic functional differential equation (SFDE) under the weaker conditions imposed in this paper, we establish a generalised existence-and-uniqueness theorem which covers a wider class of nonlinear SFDEs as demonstrated by the examples discussed in this paper. Moreover, from our improved results follow some new criteria on the stochastic asymptotic stability for SFDEs.     

分数阶微分方程多点分数阶边值问题

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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.     

几类非线性微分系统极限环的存在性和唯一性

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Nonsmooth bifurcations of equilibria in planar continuous systems

In this paper we present a procedure to find all limit sets near bifurcating equilibria in a class of hybrid systems described by continuous, piecewise smooth differential equations. For this purpose, the dynamics near the bifurcating equilibrium is locally approximated as a piecewise affine systems defined on a conic partition of the plane. To guarantee that all limit sets are identified, conditions for the existence or absence of limit cycles are presented. Combining these results with the study of return maps, a procedure is presented for a local bifurcation analysis of bifurcating equilibria in continuous, piecewise smooth systems. With this procedure, all limit sets that are created or destroyed by the bifurcation are identified in a computationally feasible manner.     

非混合单调二元算子方程(组)解的存在唯一性

利用锥理论及Banach压缩映射原理,在不要求上、下解条件及算子紧性与连续性的条件下,建立了一类满足更一般序关系条件的非混合单调二元算子方程组(?)解的存在唯一性定理,以及非单调二元算子方程T(x,x)=x和非单调一元算子方程Lx=x解的存在唯一性定理,推广了最近相关文献的研究结果.     

关于一类平面n+2次系统极限环唯一性的一个注记

本文继续完善文[1]和[2]的工作,利用广义Lienard方程和张芷芬唯一性定 理证明了,当n≥3时一类n+2次生化反应系统极限环的唯一性.至此,该系统极 限环唯一性问题得到完整解决.     

QUALITATIVE ANALYSIS OF A MULTIMOLECULEBIOCHEMICAL SYSTEM

ＱＵＡＬＩＴＡＴＩＶＥＡＮＡＬＹＳＩＳＯＦＡＭＵＬＴＩＭＯＬＥＣＵＬＥＢＩＯＣＨＥＭＩＣＡＬＳＹＳＴＥＭＧｕＤａｏｘｉｕ（ＨｕｎａｎＥｄｕｃａｔｉｏｎａｌｉｎｓｔｉｔｕｔｅ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒｔｈｅａｕｔｈｏｒｐｒｏｖｅｓｓｏｍ...     

关于方程 +φ(y)+Ψ(y)η(y)=0的极限环问题

本文研究了(1)的极限环问题,给出了判定其有无闭轨及极限环唯一的若干准则。推广和改进了[1]的结果。     

混合单调算子的不动点存在唯一性定理及其应用

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