具有间隙的动力系统的极限环

本文研究了系统ｘ＋ｆ（ｘ）＋ｇ（ｘ）＝０的极限环的存在性，其中ｇ（ｘ）有两个间断点并且不满足ｇ（ｘ）·ｘ＞０（ｘ≠０）．还引进［２，３］中定义的Филиппов解的概念，利用Филиппов解的整体存在性和普遍唯一性定理解决方程的解的存在唯一性问题，得到了几个极限环存在性定理     

状态反馈脉冲控制Holling-Tanner系统周期解的存在性

该文研究具有状态反馈脉冲控制的Holling-Tanner系统.在连续系统有唯一极限环及正平衡点为不稳定的焦点的前提下,利用微分方程几何理论、后继函数及数学分析的方法,获得脉冲系统阶1周期解的存在性、唯一性及轨道稳定性的充分条件.利用数值模拟验证主要结论,并且数值结果显示对某些参数在连续系统的极限环内存在脉冲系统的阶k周期解.     

一类Leslie模型的定性分析

对一类Leslie模型进行定性分析，研究了其极限环的存在性，不存在性和唯一性．证明了该系统在细焦点外围至多有一个极限环，以及如果系统有奇数个极限环，则它恰有一个极限环．     

一类基于等级结构的n维食物链系统最优收获

考虑一类具年龄等级结构的n维食物链种群系统的最优收获问题,首先利用压缩映射定理,研究系统解的适定性;其次构造极值化序列和运用相关的紧性定理证明控制问题最优解的存在性;最后通过构造共轭系统和利用法锥的概念刻画,得出最优收获问题最优解的一阶必要条件.     

一类摄动系统的Poincaré分叉与极限环

本文利用临界情形的隐函数存在定理讨论了一类摄动系统分支周期解的存在性与稳定性,利用后继函数法讨论了该系统极限环的存在性、唯一性和稳定性。     

非线性微分方程零解的全局吸引性与包围高阶奇点的极限环的存在性

非线性微分方程零解的全局吸引性与包围高阶奇点的极限环的存在性王向荣，冯滨鲁，韩茂安（山东矿业学院数学系泰安２７１０１９）关键词：全局吸引性；极限环；存在性ＡＭＳ（１９９１）主题分类：３４Ｃ０５．考虑系统假设ｈ，Ｆ，ｇ均连续且满足解的存在唯一性条件，且...     

一类摄动系统的Poincare分叉与极限环

本文利用临界情形的隐函数存在定理讨论了一类摄动系统分支周期解的存在性与稳定性，利用后继函数法讨论了该系统极限环的存在性、唯一性和稳定性。     

一类具年龄结构n维食物链模型的最优收获控制

研究一类具有年龄结构n维食物链模型的最优收获控制.利用不动点定理,证明了系统非负解的存在性和唯一性.由Mazur定理,证明了最优控制策略的存在性,同时由法锥概念的特征刻画,还得到了控制问题最优解存在的必要条件.     

一个三维Chemostat竞争系统的Hopf分支和周期解

本文研究了一个三维Chemostat竞争系统的解的结构,分析了平衡点的稳定性和当系统的某一微生物物种处于竞争劣势趋于灭绝时另一微生物物种和养料的二维流形上极限环的存在性,以及系统的Hopf分支问题.文中用Friedrich方法得到了系统存在Hopf分支的条件,并判定了周期解的稳定性.     

状态反馈脉冲控制比率依赖Holling-Tanner系统的周期解

本文研究具有状态反馈脉冲控制的比率依赖Holling-Tanner系统.在连续系统的正平衡点为不稳定焦点的前提下,利用微分方程几何理论及后继函数方法,获得脉冲系统阶1周期解的存在性、唯一性及轨道稳定性.利用数值模拟验证主要结论,并且数值结果得到在极限环内脉冲系统存在阶k周期解.最后,给出主要结论.     

Some Conditions for the Nonexistence of Limit Cycles in a Predator-Prey System

In this paper, we study the problem of the existence of limit cycles for a predator-prey system with a functional response. It is assumed that the functional response is positive, increasing, concave down, and its third derivative has a unique root. A necessary condition for the nonexistence of limit cycles is presented. Some conditions are given under which the necessary condition is also the sufficient condition for the nonexistence of limit cycles.     

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.     

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.     

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.     

Existence of a polycycle in non-Lipschitz Gause-type predator-prey models

We study Gause-type predator-prey models when the interaction between predator and prey is not locally-Lipschitz continuous in the absence of one of them. We shall show that in this case there appears a polycycle, which affects the existence of limit cycles for the system. We apply the results to study the existence of limit cycles for a classical Gause system.     

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.     

Limit cycles in a generalized Gause-type predator–prey model

We study the problem of the existence of limit cycles for a generalized Gause-type predator–prey model with functional and numerical responses that satisfy some general assumptions. These assumptions describe the effect of prey density on the consumption and reproduction rates of predator. The model is analyzed for the situation in which the conversion efficiency of prey into new predators increases as prey abundance increases. A necessary and sufficient condition for the existence of limit cycles is given. It is shown that the existence of a limit cycle is equivalent to the instability of the unique positive critical point of the model. The results can be applied to the analysis of many models appearing in the ecological literature for predator–prey systems. Some ecological models are given to illustrate the results.     

Existence of an optimal control for a coupled FBSDE with a non degenerate diffusion coefficient

We consider a controlled system driven by a coupled forward–backward stochastic differential equation with a non degenerate diffusion matrix. The cost functional is defined by the solution of the controlled backward stochastic differential equation, at the initial time. Our goal is to find an optimal control which minimizes the cost functional. The method consists to construct a sequence of approximating controlled systems for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we establish the existence of a relaxed optimal control to the initial problem. The existence of a strict control follows from the Filippov convexity condition.     

Conditions for the Existence of Two Limit Cycles in a System with Hysteresis Nonlinearity

This work deals with a two-dimensional automatic control system containing a single nonlinear hysteretic element in the general form. The conditions sufficient for the existence of at least two limit cycles in the system are presented. To prove the existence of cycles, three closed contours embedded into each other are constructed on the phase manifold by “sewing” together pieces of the level lines of various Lyapunov functions. System trajectories cross the inner contour “from outside inwards” and the middle contour “from inside outwards.” The outer contour is crossed by system trajectories “from outside inwards.” The existence of these contours proves the presence of at least two limit cycles in the system. This paper is a continuation of our earlier published work “Conditions for the Global Stability of a Single System with Hysteresis Nonlinearity,” in which the conditions of global stability in this system are formulated.     

Boundary control of non-well-posed elliptic equations with nonlinear Neumann boundary conditions

In this paper, we consider a boundary control problem governed by a class of non-well-posed elliptic equations with nonlinear Neumann boundary conditions. First, the existence of optimal pairs is proved. Then by considering a well-posed penalization problem and taking limit in the optimality system for penalization problem, we obtain the necessary optimality conditions for optimal pairs of initial control problem.