Multiple limit cycle bifurcations of the FitzHugh–Nagumo neuronal model

In this paper, we complete the global qualitative analysis of the well-known FitzHugh–Nagumo neuronal model. In particular, studying global limit cycle bifurcations and applying the Wintner–Perko termination principle for multiple limit cycles, we prove that the corresponding dynamical system has at most two limit cycles.     

Erratum to “Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey” [Appl. Math. Modell. (2011) 366-381]

The aim of this paper is to correct two mistakes in [Appl. Math. Modell. (2011) 366-381], which are: the function defining the time rescaling given and the inclusion of a parameter outside of model.For a modified Leslie-Gower type predator-prey model considering the Allee effect on prey, a change of variables and a new time rescaling generating a diffeomorphism is proved; a topologically equivalent system to the original one is obtained, which is the same studied in the mentioned paper; we claim that the results and conclusions obtained are correct and the errors have not further implications.     

三次系统存在一类双纽线分界线环充要条件

给出了中心对称三次系统存在一类双纽线分界线环的充要条件，并举出此系统至少还存在四个极限环的（2．2）分布的例子．还举出了中心对称三次系统至少存在六个极限环作（3.3）分布以及五个极限环，其中一个极限环包围作（2．2）分布的四个极限环的例子．     

On the unique mapping relationship between initial and final quantum states

In its standard formulation, quantum mechanics presents a very serious inconvenience: given a quantum system, there is no possibility at all to unambiguously (causally) connect a particular feature of its final state with some specific section of its initial state. This constitutes a practical limitation, for example, in numerical analyses of quantum systems, which often make necessary the use of some extra assistance from classical methodologies. Here it is shown how the Bohmian formulation of quantum mechanics removes the ambiguity of quantum mechanics, providing a consistent and clear answer to such a question without abandoning the quantum framework. More specifically, this formulation allows to define probability tubes, along which the enclosed probability keeps constant in time all the way through as the system evolves in configuration space. These tubes have the interesting property that once their boundary is defined at a given time, they are uniquely defined at any time. As a consequence, it is possible to determine final restricted (or partial) probabilities directly from localized sets of (Bohmian) initial conditions on the system initial state. Here, these facts are illustrated by means of two simple yet physically insightful numerical examples: tunneling transmission and grating diffraction.     

一个群体防卫捕-食系统的定性分析

该文对一个群体防卫捕一食系统进行了较全面的定性分析．讨论了分界线的相对位置，得到了极限环的存在性、唯一性以及分界线环的存在性，首次证明了群体防卫捕一食系统可以至少存在两个或三个极限环．     

On an integrable discretization of integrable systems with a separatrix

An integrable time-discretization of integrable Hamiltonian systems with a separatrix is considered being based on Hirota's bilinear formalism. It is proved that a discrete-time simple pendulum has a complete set of exact solutions, a conserved quantity and a separatrix. The value of the conserved quantity which characterizes the separatrix is remarkably congruent with the value of the continuous-time Hamiltonian. A discrete-time anharmonic oscillator is also shown to have the same property.     

Jump of the adiabatic invariant at a separatrix crossing: Degenerate cases

An expression for the quasi-random jump of the adiabatic invariant at a separatrix crossing is obtained for a slow-fast Hamiltonian system with two degrees of freedom in the case when the separatrix passes through a degenerate saddle point in the phase plane of the fast variables. The general case with an arbitrary degree of degeneracy was considered, and this degree is assumed to remain fixed in the process of evolution of the slow variables. The typical value of the jump is larger than in the non-degenerate case studied earlier. Though strongly degenerate, such a setting can be relevant for physical problems. The influence of the asymmetry of a phase portrait on the magnitude of adiabatic invariant jumps was considered as well. An example of this kind is studied, namely the motion of ions in current sheets with complex inner structure.     

磁重联X线附近区域的等离子体结构分布及其演化

高能电子是磁重联中产生的重要现象,卫星Wind和Cluster在磁重联的多个区域都观测到了高能电子。针对X型磁重联,结合粒子动力学方程的部分解析表达式和Hall MHD计算机数值模拟结果,研究了X线附近区域的电子和离子运动的分离特性,得到了该区域垂直重联平面的磁场分量和速度分量随时间的演化特征。研究结果表明,磁重联过程中起支配作用的电子的运动是高度结构化的,受多种因素作用,其运动速度呈现非线性变化的特点,在垂直重联平面z方向的磁场分量和速度分量都存在拐点,拐点前后磁重联的电场分量的波动特性有明显差异。     

The stochastic web on a spherical surface generated by simple, 3-dimensional rotational transformations

It is shown that the stochastic web (or chaotic web) on the surface of a sphere can be generated by a simple, 3-dimentional rotational map, constructed by three rotational angles about each coordinate axis: x-axis, y-axis and z-axis. It is remarkable that the rotational angles in our model do not need to be complicated functions of the coordinates. As a matter of fact, the stochastic web is found when our map only consists of one simple functional rotational angle and two constant rotational angles, under certain resonance conditions. The trajectories are computed and the 3-dimentional plots of the stochastic web on the spherical surface are also presented.     

具收获项的Ⅱ类功能反应的捕-食系统的定性分析~Ⅰ

在文[7][8]的基础上，分析一类食饵具常数收获项的Ⅱ类功能反应的捕食系统．通过对两鞍点分界线相互位置仔细的分析，得到了比较完备的定性结论，特别是第一次在生态系统的讨论中得到了三个极限环的存在性．