Limit cycles of the two-dimensional motion field

Recently, it has been shown that there can be limit cycles in the vector field generated by the perspective projection on the image plane of the three dimensional velocity field of a certain class of non-planar rotating surfaces. In this paper, it is shown that, for any possible rigid motion, there cannot be limit cycles in the motion field of a planar surface. Therefore, the presence of limit cycles in the motion field is necessarily due to the non-planar structure of the viewed scene. An experiment on real images is also presented in which a limit cycle occurs when two planar patches have different orientation in space rotate around a fixed axis.     

Stable periodic solutions for two species,density dependent coevolution

This paper studies a four dimensional system of time-autonomous ordinary differential equations which models the interaction of two diploid, diallelic populations with overlapping generations. The variables are two population densities and an allele frequency in each of the populations. For single species models, the existence of periodic solutions requires that the genotype fitness functions be both frequency and density dependent. But, for two species exhibiting a predator-prey interaction, two examples are presented where there exists asymptotically stable cycles with fitness functions only density dependent. In the first example, the Hopf bifurcation theorem is used on a two parameter, polynomial vector field. The second example has a Michaelis-Menten or Holling term for the interaction between predator and prey; and, for this example, the existence and uniqueness of limit cycles for a wide range of parameter values has been established in the literature.     

一类两种群均有收获率捕食系统的细焦点与极限环

研究一类具有HollingⅡ类功能反应且两种群均为非常数收获率的捕食系统,其中食饵种群具有非线性密度制约.利用微分方程定性与稳定性理论及分支理论,得到系统平衡点的性态及极限环存在与否的充分条件,利用Hopf分支理论得到存到多个极限环的充分条件.     

Homoclinic and heteroclinic orbits to a cycle in a tri-trophic food chain

 The asymptotic behavior of a tri-trophic food chain model is studied. The analysis is carried out numerically, by finding both local and global bifurcations of equilibria and limit cycles. The existence of transversal homoclinic orbits to a limit cycle is shown. The appearance of homoclinic orbits, by moving through a homoclinic bifurcation point, is associated with the sudden disappearance of a chaotic attractor. A homoclinic bifurcation curve, which bounds a region of extinction, is continued through a two-dimensional parameter space. Heteroclinic orbits from an equilibrium to a limit cycle are computed. The existence of these heteroclinic orbits has important consequences on the domains of attraction. Continuation of non-transversal heteroclinic orbits through parameter space shows the existence of two codimension-two bifurcations points, where the saddle cycle is non-hyperbolic. The results are summarized by dividing the parameter space in subregions with different asymptotic behavior. Received: 25 February 1998 / Revised version: 19 August 1998     

Small autocatalytic reaction networks—III. Monotone growth functions

The classification of the dynamical behaviour of first order replicator equations is extended to models with monotonical growth rates. It is shown that for two species there is a general classification independent of the particular form of the growth function. For three species a common dynamical behaviour for all power laws can be found and the existence of limit cycles is disproved. For more general growth functions, however, limit cycles may occur.     

一类生化系统的稳定性

本文研究一类生化系统的极限环的存在性与唯一性;分析了系统轨线的全局结构,指出了系统在无穷远点处的轨线存在奇异方向;解释了极限环消失的原因．     

A tale of two cycles – distinguishing quasi-cycles and limit cycles in finite predator–prey populations

Periodic predator – prey dynamics in constant environments are usually taken as indicative of deterministic limit cycles. It is known, however, that demographic stochasticity in finite populations can also give rise to regular population cycles, even when the corresponding deterministic models predict a stable equilibrium. Specifically, such quasi-cycles are expected in stochastic versions of deterministic models exhibiting equilibrium dynamics with weakly damped oscillations. The existence of quasi-cycles substantially expands the scope for natural patterns of periodic population oscillations caused by ecological interactions, thereby complicating the conclusive interpretation of such patterns. Here we show how to distinguish between quasi-cycles and noisy limit cycles based on observing changing population sizes in predator – prey populations. We start by confirming that both types of cycle can occur in the individual-based version of a widely used class of deterministic predator – prey model. We then show that it is feasible and straightforward to accurately distinguish between the two types of cycle through the combined analysis of autocorrelations and marginal distributions of population sizes. Finally, by confronting these results with real ecological time series, we demonstrate that by using our methods even short and imperfect time series allow quasi-cycles and limit cycles to be distinguished reliably.     

一类食饵种群具有常数存放率的Kolmogorov系统的定性分析

本文对一类食饵种群具有常数存放率的Kolmogorov系统进行了定性分析,研究了系统平衡点的性态,证明了极限环的存在和唯一性．     

一类食饵种群具有常数收获率的Kolmogolov系统的非线性分析

本文对一类食饵种群具有常数收获率的Ｋｏｌｍｏｇｏｌｏｖ系统进行了非线性分析,研究了 系统平衡点的性态,在较宽的条件下证明了系统不存在闭轨线;并获得了系统存在两个极限环的条件。     

具有抑制剂的恒化器竞争模型的分析（英文）

考虑了具有抑制剂的两种群竞争一种微生物的恒化器模型,其中吸收函数和功能反应函数都是营养的一般单调递增函数,并且一种微生物能够分泌一种对另一种微生物起致命影响的抑制剂.利用常微分方程定性理论,首先得到了平衡点的存在条件和局部渐进稳定性;然后讨论了全局渐进稳定性,以及极限环和Hopf分支的存在性.     

A Qualitative Analysis of Quartic Ecology Differential System

Thequarticecologydifferentialsystemwasstudied,Sometheoremsofexistenceanduniquenessoflimitcycleswereobtained.     

Passive bipedal walking with phasic muscle contraction

The existence of self-organizing walking patterns is often considered the result of a mechanical system interacting with the environment and a (neural) oscillating unit. The pattern generators might be thought of as an indispensable component for the existence of limit cycle behavior. This paper shows that this is not a necessity for the existence of a self-organizing bipedal walking pattern. Stable walking cycles emerge from a simple passive bipedal structure, with an energy source inevitably present to sustain the oscillation. In this work the energy source is chosen to be phasic muscle contraction. A two-dimensional model is composed of two legs and a hip mass, symbolizing the trunk. The stance leg stiffness is generated by two muscles. The hip stiffness is generated by four muscles. Muscle activation is caused by two reflex-like trigger signals, without feedback control. Human equivalent model parameters such as geometry and mass distribution were assumed. With return map analysis, the model is analyzed on periodic behavior. Stable walking cycles were found and could be manipulated during walking by varying the muscle or reflex parameters, forcing the oscillation to converge to a new attractor. Received: 5 November 1998 / Accepted in revised form: 26 March 1999     

一类微生物连续培养竞争模型的定性分析

研究一类微生物连续培养竞争模型的解的结构,分析了平衡点的稳定性及平衡点附近极限环存在唯一性,证明系统存在正向不变集。     

一类SIRS流行病模型可至少存在两个极限环

本文研究了一类ＳＩＲＳ流行病模型，主要得到了在一定条件下可至少存在两个极限环的结论．     

一类生化系统的极限环

本文研究了一类生化系统dx/dt=b-cx x^py,dy/dt=a-x^py,a,b,c,p均为正实数,得到了该系统不存在极限环,以及存在唯一稳定的极限环的充要条件。     

一类非线性微分动力系统的定性分析

本文研究了一类非线性微分动力系统０,ｂ＞０,Ｐ＞０）的定性行为,完整地解决了系统的极限环的不存在性、存在性和唯一性问题。得到系统有唯一极限环当且仅当（Ｐ一１）ａ－ｂ＞（ａ＋ｂ）~（Ｐ＋１）     

一类捕食—食饵动力系统的定性分析

本文对一类形式广泛的捕食-食饵动力系统进行了定性研究，获得了确保此生态系统的所有解有界及极限环存在与唯一的若干充分条件，并给出了所获结论在生态学上的意义。     

一类多分子生化反应系统的极限环

应用微分方程定性理论,研究了生化反应中一类多分子一级饱和反应的数学模型的极限环的存在性、不存在性和唯一性问题．     

一类一级饱和反应系统的极限环

本文研究生化反应中一类饱和反应的数学模型：应用微分方程定性理论,完整地解决了该系统极限环的存在性、唯一性和不存在性等问题．     

可逆两分子饱和反应动力系统的极限环

研究生化反应中一类可逆两分子饱和反应的数学模型dx/dt＝δ-xy cy^2,dy/dt=xy-cy^2-ay/b b,应用微分方程定性理论,完整地解决了该系统极限环的存在性,不存在性和唯一性。