一类具功能反应的食饵-捕食者系统的定性分析

运用微分方程定性与稳定性分析的方法研究了一类食饵-捕食者系统的定性行为,讨论了该系统的平衡点的性态,给出了该系统极限环不存在性存在性的充分条件以及至多存在一个极限环的充分条件,从而补充和推广了前人已有的结果.     

关于一类平面三次系统的极限环

关于系统(1)的极限环的存在性问题,[1,2]已有过论述,[1]指出,当系统(1)仅有一个初等奇点,F(x,y)=0表示椭圆,且原点位于其内部时,系统(1)存在极限环;[2]考虑系统(1)有一个以上初等奇点,F(x,y)=0表示椭圆时的情况,给出系统(1)存在极限环的充分条件.本文在[1,2]的基础上继续研究系统(1)的极限环的存在性问题,与[1,2]不同,本文不但考虑 F(x,y)=0表示椭圆时情况,而且还考虑了 F(x,y)=0表示其它二次曲线时的情况,不但考虑了系统(1)有初等奇点时情况,而且还考虑了系统(1)有高次奇点时情况,给出系统(1)极限环存在的充分条件.     

具有椭圆解的系统(E_3~2)的极限环的存在性

本文讨论了一类具有椭圆解的三次系统(E_3~2),证明了当椭圆解为此系统的极限环时,还可以存在其它极限环,并描绘出当具有椭圆极限环时此系统的所有可能的全局相图,此外,还举出了一个以此椭圆为无返回映射分界线环的例子,其内部包含三个奇点和至少一个极限环.     

一类具有二阶细焦点的二次系统

文[2]已经证明,具有三阶细焦点的二次系统(叶彦谦形式)当n=0时不存在极限环。本文继续运用文[2]的方法,得到了具有二阶细焦点的二次系统当n=0时在二阶细焦点外围存在极限环的条件和不存在极限环的条件,同时证明这种系统在其他奇点外围不存在极限环。     

一类带有三次项的平面五次微分系统的全局拓扑结构分析

一类带有三次项的平面五次微分系统在Poincare变换下可以讨论系统的无穷远奇点的性质,进而得到奇点附近轨线的拓扑结构,并利用判断函数给出极限环存在与否的条件,补充完善了五次系统的定性分析.     

具有椭圆解的系统(E_3~2)的极限环的存在性

本文讨论了一类具有椭圆解的三次系统(E_3~2),证明了当椭圆解为此系统的极限环时,还可以存在其它极限环,并描绘出当具有椭圆极限环时此系统的所有可能的全局相图,此外,还举出了一个以此椭圆为无返回映射分界线环的例子,其内部包含三个奇点和至少一个极限环。     

一类三次系统的极限环

主要研究一类三次系统的极限环存在性问题,推广了C.Chicone[2]的结果,给出此类系统极限环存在定理.     

一类具细焦点的三次系统极限环的唯一性

继续相关文献的工作,给出与二次系统Ⅰ相伴的一类三次系统在奇点N(0,1/n)的焦点量公式,证明了系统在细焦点N外围至多有一个极限环,同时证明了当N或O为细焦点时,系统在另一个焦点外围无极限环,结合相关文献的结论,说明了具有细焦点的该系统在全平面至多有一个极限环.     

Lotka-Volterra系统与Kolmogorov系统极限环的存在性与中心焦点的算法化判定[英文]

本文得到Kolmogorov系统为中心的充分条件.对三维Lotka-Volterra系统和Kolmogorov系统之间的关系进行探讨.通过研究Kolmogorov系统极限环的存在性,得到三维Lotka-Volterra系统极限环的存在性.     

Lienard方程极限环的存在性

本文讨论了Lienard方程极限环的存在性,改进并补充了篇末文献中的有关结果.     

Trajectory tracking control of variable length pendulum by partial energy shaping

This paper concerns a trajectory tracking control problem for a pendulum with variable length, which is an underactuated mechanical system of two degrees-of-freedom with a single input of adjusting the length of the pendulum. We aim to study whether it is possible to design a time-invariant control law to pump appropriate energy into the variable length pendulum for achieving a desired swing motion (trajectory) with given desired energy and length of the pendulum. First, we show that it is difficult to avoid singular points in the controller designed by using the conventional energy-based control approach in which the total mechanical energy of the pendulum is controlled. Second, we present a tracking controller free of singular points by using only the kinetic energy of rotation and the potential energy of the pendulum and not using the kinetic energy of the motion along the rod. Third, we analyze globally the motion of the pendulum and clarify the stability issue of two closed-loop equilibrium points; and we also provide some conditions on control parameters for achieving the tracking objective. Finally, we show numerical simulation results to validate the presented theoretical results.     

On the existence of a stable limit cycle to a piecewise linear system in R2

The present paper is devoted to the existence of limit cycles of planar piecewise linear (PWL) systems with two zones separated by a straight line and singularity of type “focus-focus” and “focus-center.” Our investigation is a supplement to the classification of Freire et al concerning the existence and number of the limit cycles depending on certain parameters. To prove existence of a stable limit cycle in the case “focus-center,” we use a pure geometric approach. In the case “focus-focus,” we prove existence of a special configuration of five parameters leading to the existence of a unique stable limit cycle, whose period can be found by solving a transcendent equation. An estimate of this period is obtained. We apply this theory on a two-dimensional system describing the qualitative behavior of a two-dimensional excitable membrane model.     

基于Labview的一级环形倒立摆逆系统轨迹法自动摆起控制

利用求解非线性方程两点边值问题,得到逆系统参考轨迹;通过设计逆系统前馈控制及LQR反馈控制器,对参考轨迹实时跟踪。基于QNET 2.0旋转倒立摆实验平台,采用Labview编程软件,实现了一级环形倒立摆的自动摆起与稳定的实物控制。实验结果证明控制算法的有效性,可以使一级环形倒立摆在一个摆周期内自动摆起,并保持稳定状态。与能量法自动起摆实验比较,逆系统算法具有较好的快速性和稳定性。     

THE E_3~1 TYPE OF CUBIC SYSTEMS WITH TWO INTEGRAL STRAIGHT LINES

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Bifurcation of periodic orbits in a class of planar Filippov systems

In this paper we discuss the perturbations of a general planar Filippov system with exactly one switching line. When the system has a limit cycle, we give a condition for its persistence; when the system has an annulus of periodic orbits, we give a condition under which limit cycles are bifurcated from the annulus. We also further discuss the stability and bifurcations of a nonhyperbolic limit cycle. When the system has an annulus of periodic orbits, we show via an example how the number of limit cycles bifurcated from the annulus is affected by the switching.     

Truncated fractal basin boundaries in the pendulum with nonperiodic forcing

Summary It is well known that oscillators such as the pendulum can have fractal basin boundaries when they are periodically forced with the consequence that the long term behavior of the system may be unpredictable. In engineering and physical applications, the forcing is often nonperiodic and eventually decays to zero, and simulation of the pendulum with decaying forcing (M. Varghese, J. S. Thorp, Physical Review Letters, vol. 60, no. 8, pp. 665–668, Feb. 1988) exhibits truncated fractal basin boundaries which also limit the system predictability. We develop a coordinate change for the pendulum with decaying forcing that allows us to apply standard qualitative methods to study the basin boundaries. We prove that the basin boundaries cannot be fractal and show by example how the extreme stretching and folding leading to a truncated fractal basin boundary may arise.     

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.     

Jacobi因子在稳定性理论中的应用

本文利用Jacobi因子的性质,得出一些有关定常系统、非定常系统以及极限环的不稳定定理.     

保守双摆的不可积性和混沌

本文用Birkhoff级数正则变换方法求出保守双摆运动方程的近似积分,并把近似积分的等值曲线与数值仿真结果作了比较.由此清楚地看出.当能级提高时,系统由近可积的成为不可积的,即其运动情况由规则的转变为混沌的.本文还介绍了演示上述性态的一个保守双摆模型.     

广义Liénard方程极限环的惟一性问题

本文证明了广义Lienard方程极限环的一个惟一性定理,并用它证明了具有 稀疏效应的捕食-食饵系统在其正奇点外围至多有一个极限环.