几类动力系统动力学性态研究
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摘要
现实的很多问题包含多种因素，这些因素的相互作用使其对应的动力系统具有丰富的动力学行为。对于多因素耦合的离散动力系统和显含空间的动力系统，相关的研究成果还很缺乏。针对当前的研究现状，本文主要做了以下工作：
1、建立了智能雷空间运动轨迹动力学模型，并给出了智能雷扫描轨迹随射程的变化曲线以及攻角和时间的变化关系。
2、研究了个体相互残杀和捕食的共同作用的离散动力系统，分析系统周期解的存在性和稳定性，并给出其生物学意义。分析了带两参数的时滞和竞争因素耦合离散动力系统，通过构造一条曲线，把参数空间分成两个区域Λ1和Λ2。在区域Λ1，系统至少有两个周期解；在曲线上，系统至少有一个周期解；在区域Λ2，系统没有周期解。
3、研究了具有空间和随机因素的动力系统，分别分析了非密度依赖和密度依赖的随机噪声和扩散的相互作用对个体空间分布和持续生存的影响。利用数学分析和数值模拟发现：第一种形式的噪声会导致个体的空间分布发生相变，即从点状斑图向线状斑图转化。随着噪声强度的增加，点状斑图的点数是先增加到最大值，而后减少。而当时间相关性很大时，噪声会诱导个体出现条状的空间分布。对于第二种形式的噪声，发现噪声会导致个体从点状分布到迷宫状的分布。对于某些噪声参数，噪声会导致个体从持续变为灭绝，并给出了个体持续和灭绝的参数区域。
4、研究了基于反应扩散方程的且具有Allee效应和非线性死亡项的动力系统，通过多尺度理论中的振幅方程分析了这两种因素的相互作用对个体空间分布的影响。研究结果表明：带有Allee效应和非线性作用项的空间动力系统具有丰富的斑图结果。随着空间参数δ的增大，斑图结构从H0向Hπ演变，经历了不同类型斑图的相变。当参数增大到某个数值，个体会呈现螺旋波式的分布。这意味着个体的分布从稳态斑图变为移动斑图，不利于个体的持续生存。
Many problems in real world contain a lot of factors and the interactions between these factors maycause rich dynamics. There is little work on discrete dynamics with multiple factors and spatial dynamics.Based on the current progress of these topics, we did the following work.
Firstly, we construct a dynamical model on the motion trail of intelligent mine and give the motiontrail and attack angle with respect to fire rate and time, respectively.
Secondly, we construct a discrete dynamical model with individuals cannibalism and predation andinvestigate the existence and stability of positive periodic solutions. Some biological meanings are shownfor the theoretical results. Furthermore, we present a discrete dynamical systems with delay and competitionand obtain the distribution of positive periodic solutions. By structuring a curve, we separate the parametersspace into two parts Λ1and Λ2. There are at least two positive periodic solutions in domain Λ1; at least onepositive periodic solution in the curve; no positive periodic solution in domain Λ2.
Thirdly, we investigate spatial dynamics with difusion and stochastic factors. We reveal the influenceof density-independent and density-dependent noise on the pattern formation and persistence of populations.By both mathematical analysis and numerical simulations, we find that density-independent noise will in-duce pattern transition from spotted pattern to stripe pattern. The number of the spotted pattern will increaseas noise intensity is small. As noise intensity increase, the number will decrease. However, when temporalcorrelation is large, stripe pattern emerges. For the spatial dynamics with density-dependent noise, noisecan induce pattern transition from spotted pattern to labyrinth pattern. What is more, noise can cause thepopulations to extinct from persistence for some values of noise intensity and temporal correlation. We alsoshow the persistence and extinction regions of populations in parameters space.
Finally, we study a spatial dynamics with Allee efect and nonlinear death rate based on reactiondifusion equation. By using amplitude equations in multiple scales, we investigate the efect of interactionsbetween these two factors on the distribution of populations. The obtained results show that spatial systemswith Allee efect and nonlinear interactions can give rise to rich dynamical behaviors. As parameter δincrease, pattern structures will change from H0to Hπ. When the parameter increases to a certain value, theindividual exhibits a spiral wave pattern formation. It means that the stationary pattern is transformed intounstable state, which is bad for the persistence of populations.
1、建立了智能雷空间运动轨迹动力学模型，并给出了智能雷扫描轨迹随射程的变化曲线以及攻角和时间的变化关系。
2、研究了个体相互残杀和捕食的共同作用的离散动力系统，分析系统周期解的存在性和稳定性，并给出其生物学意义。分析了带两参数的时滞和竞争因素耦合离散动力系统，通过构造一条曲线，把参数空间分成两个区域Λ1和Λ2。在区域Λ1，系统至少有两个周期解；在曲线上，系统至少有一个周期解；在区域Λ2，系统没有周期解。
3、研究了具有空间和随机因素的动力系统，分别分析了非密度依赖和密度依赖的随机噪声和扩散的相互作用对个体空间分布和持续生存的影响。利用数学分析和数值模拟发现：第一种形式的噪声会导致个体的空间分布发生相变，即从点状斑图向线状斑图转化。随着噪声强度的增加，点状斑图的点数是先增加到最大值，而后减少。而当时间相关性很大时，噪声会诱导个体出现条状的空间分布。对于第二种形式的噪声，发现噪声会导致个体从点状分布到迷宫状的分布。对于某些噪声参数，噪声会导致个体从持续变为灭绝，并给出了个体持续和灭绝的参数区域。
4、研究了基于反应扩散方程的且具有Allee效应和非线性死亡项的动力系统，通过多尺度理论中的振幅方程分析了这两种因素的相互作用对个体空间分布的影响。研究结果表明：带有Allee效应和非线性作用项的空间动力系统具有丰富的斑图结果。随着空间参数δ的增大，斑图结构从H0向Hπ演变，经历了不同类型斑图的相变。当参数增大到某个数值，个体会呈现螺旋波式的分布。这意味着个体的分布从稳态斑图变为移动斑图，不利于个体的持续生存。
Many problems in real world contain a lot of factors and the interactions between these factors maycause rich dynamics. There is little work on discrete dynamics with multiple factors and spatial dynamics.Based on the current progress of these topics, we did the following work.
Firstly, we construct a dynamical model on the motion trail of intelligent mine and give the motiontrail and attack angle with respect to fire rate and time, respectively.
Secondly, we construct a discrete dynamical model with individuals cannibalism and predation andinvestigate the existence and stability of positive periodic solutions. Some biological meanings are shownfor the theoretical results. Furthermore, we present a discrete dynamical systems with delay and competitionand obtain the distribution of positive periodic solutions. By structuring a curve, we separate the parametersspace into two parts Λ1and Λ2. There are at least two positive periodic solutions in domain Λ1; at least onepositive periodic solution in the curve; no positive periodic solution in domain Λ2.
Thirdly, we investigate spatial dynamics with difusion and stochastic factors. We reveal the influenceof density-independent and density-dependent noise on the pattern formation and persistence of populations.By both mathematical analysis and numerical simulations, we find that density-independent noise will in-duce pattern transition from spotted pattern to stripe pattern. The number of the spotted pattern will increaseas noise intensity is small. As noise intensity increase, the number will decrease. However, when temporalcorrelation is large, stripe pattern emerges. For the spatial dynamics with density-dependent noise, noisecan induce pattern transition from spotted pattern to labyrinth pattern. What is more, noise can cause thepopulations to extinct from persistence for some values of noise intensity and temporal correlation. We alsoshow the persistence and extinction regions of populations in parameters space.
Finally, we study a spatial dynamics with Allee efect and nonlinear death rate based on reactiondifusion equation. By using amplitude equations in multiple scales, we investigate the efect of interactionsbetween these two factors on the distribution of populations. The obtained results show that spatial systemswith Allee efect and nonlinear interactions can give rise to rich dynamical behaviors. As parameter δincrease, pattern structures will change from H0to Hπ. When the parameter increases to a certain value, theindividual exhibits a spiral wave pattern formation. It means that the stationary pattern is transformed intounstable state, which is bad for the persistence of populations.
引文
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