一类非自治系统概周期解的存在唯一性

研究高压输电网中出现的概周期振荡现象,结合运用 Ｌｉａｐｕｎｏｖ 函数,获得了系统产生概周期振荡的先兆性条件,为避免系统产生概周期振荡提供了参考数据·     

一类三维生态动力系统的Hopf分支

考虑一类具偏食习惯的捕食者与被捕食者模型．利用中心流形定理和 Hopf分支理论讨论并证明了该系统在一定条件下产生Hopf分支，得到中心流形、小振幅空间周期解的渐近表达式，同时给出了周期解稳定性判据．     

温盐双扩散系统对流扩散周期解的线性与非线性稳定性分析

本文对温盐双扩散系统的稳定性问题引入了一种简洁的强非线性自治系统稳定性的分析方法，用摄动理论得到了无穷小运动下线性周期解的单调与振荡分支的存在范围及有限振幅运动下非线性周期解的振荡分支在０＜ｒｓ－ｒｓｃ＜＜１条件下的存在区域及稳定性区域，给出了不同涡旋方向下的稳定性结论．     

具时滞能源价格模型的动力学性质

以滞量τ为分支参数,研究了具时滞的能源价格模型的动力学行为,这些行为包括:系统在平衡点附近的稳定性,局部Hopf分支的存在性,发生条件.Hopf分支的方向,分支周期解的稳定性以及分支随参数变化其周期解的周期变化.最后通过数值模拟验证了理论分析结果,并用分支理论解释了能源价格模型产生且维持周期振荡的原因.     

动载荷下不可压缩超弹性球形薄膜的若干定性性质

对于由横观各向同性不可压缩的Rivlin-Saunders材料组成的球形薄膜,研究了薄膜的内、外表面在周期阶梯载荷作用下的轴对称变形的非线性动力学特性．通过令球形结构的厚度趋近于1,得到了近似描述薄膜径向对称运动的二阶非线性常微分方程．详细讨论了解的定性性质．特别地,给出了球形薄膜随时间的运动产生非线性周期振动的可控性条件,证明了在某些情形下周期振动的振幅会出现“∞”型同宿轨道以及周期振动的振幅会出现不连续增长现象,并给出了相应的数值模拟．     

一类HIV病毒动力学模型的全局稳定性与周期解

建立了一类较广泛的HIV感染CD4+T细胞病毒动力学模型,给出了一个感染细胞在其整个感染期内产生的病毒的平均数(基本再生数)R0的表达式,运用Lyapunov原理和Routh-Hurwitz判据得到了该模型的未感染平衡点与感染平衡点的存在性与稳定性条件.同时也得到了模型存在轨道渐近稳定周期解和系统持续生存的条件,并通过数值模拟验证了所得到的结果.     

Bistable Periodic Traveling Wave Solutions to Lattice Competitive Systems北大核心CSCD

该文研究了一类格竞争系统的双稳周期行波解的存在性.首先,将两种群竞争系统转化为合作系统;其次,构造合作系统的上下解,并建立比较原理,得到当初始函数满足一定条件时,解在无穷远处是收敛的;最后,利用黏性消去法证明系统连接两个稳定周期平衡点的行波解的存在性.     

区间周期系统的平稳振荡

本文给出了区间周期系统平稳振荡的概念，建立了区间矩阵稳定性与区间周期系统平稳振荡之间的关系，推广和改进了文[1]的结果．     

具有时滞的细胞神经网络周期解与稳定性

本文研究了具有时滞的细胞神经网络周期解存在性和平凡解的稳定性问题。利用Ｌｙａｐｕｎｏｖ函数法并结合不等式分析技巧,我们首先证明了时滞细胞网络的解是有界的,然后建立了时滞细胞神经物周期解的存在准则,最后在时滞细胞神经网络有平衡点时,给出了神经网络系统的平衡点指数稳定的充分条件。其结果推广了文「７,８」的相应结果。     

食饵为Smith增长且具有合作狩猎的捕食模型动力学分析

建立了一类食饵种群为Smith增长并且考虑捕食者合作狩猎的捕食与被捕食模型,通过研究发现捕食者合作狩猎强度和食饵的净增长率会影响种群的共存状态.并且给出系统存在一个或多个共存平衡点的条件,当出现两个共存平衡点时,系统会呈现双稳状态,即种群或者保持稳定共存,或者捕食种群灭绝,食饵种群达到饱和;并且系统会在某些平衡点处发生Hopf分支,产生持续捕食者-食饵振荡;当两个共存平衡点重合时,系统会发生BT分支,呈现单稳状态,捕食者灭绝平衡成为惟一稳定状态.同时进行了相应的数值模拟和生物解释.     

Effects of feedback regulation on vegetation patterns in semi-arid environments

It is well known that vegetation patterns characterize the distribution of the vegetation and provide some signs for vegetation protection. The positive feedbacks regulation between the water and plant biomass play an important role in the vegetation patterns in semi-arid environments, yet its influence on vegetation patterns is far from being well understood. In order to reveal a mechanism of positive feedbacks on pattern formation, a water-biomass model in semi-arid environments with soil-water diffusion feedback is presented. Our results reveal that, as the soil-water diffusion intensity decreases, the pattern transitions: gap patterns → stripe (labyrinth) patterns → spot patterns emerge. More importantly, when the soil-water diffusion feedback intensity is smaller, the feedback will promote the growth of the vegetation; when the feedback intensity is much larger, the vegetation biomass will decrease and the feedback may induce the emergence of desertification. Additionally, the rainfall can also induce the pattern transition. As the rainfall capacity decreases, the vegetation disappears and becomes the bare soil state. Our findings highlight the relationship among feedback intensity, rainfall and pattern dynamics of the vegetation.     

Turing vegetation patterns in a generalized hyperbolic Klausmeier model

The formation of Turing vegetation patterns in flat arid environments is investigated in the framework of a generalized version of the hyperbolic Klausmeier model. The extensions here considered involve, on the one hand, the strength of the rate at which rainfall water enters into the soil and, on the other hand, the functional dependence of the inertial times on vegetation biomass and soil water. The study aims at elucidating how the inclusion of these generalized quantities affects the onset of bifurcation of supercritical Turing patterns as well as the transient dynamics observed from an uniformly vegetated state towards a patterned state. To achieve these goals, linear and multiple-scales weakly nonlinear stability analysis are addressed, this latter being inspected in both large and small spatial domains. Analytical results are then corroborated by numerical simulations, which also serve to describe more deeply the spatio-temporal evolution of the emerging patterns as well as to characterize the different timescales involved in vegetation dynamics.     

Rise and Fall of Periodic Patterns for a Generalized Klausmeier–Gray–Scott Model

In this paper we introduce a conceptual model for vegetation patterns that generalizes the Klausmeier model for semi-arid ecosystems on a sloped terrain (Klausmeier in Science 284:1826–1828, 1999). Our model not only incorporates downhill flow, but also linear or nonlinear diffusion for the water component. To relate the model to observations and simulations in ecology, we first consider the onset of pattern formation through a Turing or a Turing–Hopf bifurcation. We perform a Ginzburg–Landau analysis to study the weakly nonlinear evolution of small amplitude patterns and we show that the Turing/Turing–Hopf bifurcation is supercritical under realistic circumstances. In the second part we numerically construct Busse balloons to further follow the family of stable spatially periodic (vegetation) patterns. We find that destabilization (and thus desertification) can be caused by three different mechanisms: fold, Hopf and sideband instability, and show that the Hopf instability can no longer occur when the gradient of the domain is above a certain threshold. We encounter a number of intriguing phenomena, such as a ‘Hopf dance’ and a fine structure of sideband instabilities. Finally, we conclude that there exists no decisive qualitative difference between the Busse balloons for the model with standard diffusion and the Busse balloons for the model with nonlinear diffusion.     

Analysis of spatial patterns in a vegetation model

In this paper, a vegetation model with spatial diffusion is investigated. By linear analysis, we give the critical values of the wavenumber of the emerging patterns. Moreover, through a detailed analysis in two spatial dimensions, the pattern formation mechanisms of steady state patterns are obtained dependent on the quantity of rainfall. The mechanisms of pattern formation suggest that rainfall plays an important role in the formation of vegetation.     

Near-resonace transverse oscillations in an elastic layer. Steady-state solutions

The solutions of the equations of the non-linear evolution of transverse oscillations in a layer of an incompressible elastic medium under conditions close to resonance conditions are investigated qualitatively and using analytical methods. The oscillations are created by a small periodic motion of one of the boundaries in its plane, with a period that is close to the period of the natural oscillations of the layer. It is assumed that the medium can possess slight anisotropy and that the amplitude of the oscillations which arise is small. Previously obtained differential equations are used, which describe the slow evolution of the wave pattern of non-linear transverse waves. Two possible formulations of problems for these equations are considered. In the first formulation, it is determined what the external action must be in order that the non-linear evolution of oscillations or periodic oscillations occurs according to a (previously specified) desired law. In the second formulation it is assumed that the periodic motion of one of the boundaries is given. It is shown that a steady-state solution, that does not vary from period to period, can be represented by a continuous solution and also by a solution which contains discontinuities in the strain and velocity components. The mechanism of the overturn of a non-linear wave during its evolution and the formation of a discontinuity are qualitatively described.     

组合超弹性球体中空穴的动态生成

根据有限变形动力学理论,研究了一个不可压超弹性材料组合球体在突加表面均布拉伸载荷作用下空穴的动态生成问题．当外加载荷超过其临界值时,除一个平凡解外,还有一个包含着球体内部生成的空穴的分叉解;证明空穴随时间的演化是周期性的非线性振动;同时给出了空穴生成时的临界载荷值、空穴振动的相图、振幅及近似的周期．     

Envelope Solitary Waves and Periodic Waves in the AB Equations

This article deals with the envelope solitary waves and periodic waves in the AB equations that serve as model equations describing marginally unstable baroclinic wave packets in geophysical fluids and also ultra‐short pulses in nonlinear optics. An envelope solitary wave has a width proportional to its velocity and inversely proportional to its amplitude. The velocity of the envelope solitary wave is partially dependent on its amplitude in the sense that the amplitude determines the upper or lower limit of the velocity. When two envelope solitary waves collide, they survive the collision and retain their identities except for a shift in the positions of both the envelopes and the carrier waves. The periodic wave solutions in sine wave form may be stable or unstable depending upon the wave parameters. When the sine wave is destabilized by small perturbations, its long‐time evolution shows a Fermi–Pasta–Ulam‐type oscillation.     

A Holling II functional response food chain model with impulsive perturbations

In this paper, we investigate a three trophic level food chain system with Holling II functional responses and periodic constant impulsive perturbations of top predator. Conditions for extinction of predator as a pest are given. By using the Floquet theory of impulsive equation and small amplitude perturbation skills, we consider the local stability of predator eradication periodic solution. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows the rich dynamics (for example: period doubling, period halfing, chaos crisis) in the positive octant. The dynamics behavior is found to be very sensitive to the parameter values and initial value.     

Stability and persistence of two-dimensional patterns

The canonical equations for evolution of the amplitude order parameters order parameters describing the nonlinear development and persistence of two-dimensional three-mode spatial patterns generated by Turing instability in dissipative systems are considered. The stability conditions for persistent hexagonal patterns are generalized, and the conditions under which patterns are either disrupted, exhibit bounded quasiperiodic or chaotic behavior, or decay under nonlinear evolution are derived. These conditions are applied to the specific three-mode amplitude evolution equations derived for the Schnakenberg model and a delay predator system in Chapter 3. Numerical results are presented for the persistence, disruption and decay of patterns in these systems, including fairly detailed comparisons with simulations results for the Snackenberg model.