竞争系统长期动力学性态的研究

发布时间：2018-06-06 19:31

  本文选题：竞争动力系统 + 离散竞争映射　； 参考：《上海师范大学》2016年博士论文

【摘要】：本文主要研究由竞争映射诱导的离散动力系统、时间回复的非自治系统和自治竞争系统的长期动力学性态,主要研究工作包括如下三方面:I.对对于一一般的竞竞争映映射建建立了了负载载单形形的存存在性性理论论及指指标公公式,并并由此根根据边边界不不动点点的局局部动动力学学性态态定义义了等等价关关系,对对三维维的Leslie/Gower映映射及Atkinson/Allen映映射给出了等价分类.首先,我们证明了一个十分容易验证的竞争映射的负载单形存在唯一性定理,由此证明了一大类的任意维离散竞争系统都存在负载单形,特别地,首次证明了任意维的Leslie/Gomer映射与Atkinson/Allen映射等无条件存在负载单形.基于负载单形存在性理论,我们给出了3-维竞争映射负载单形上的指标和公式并研究了负载单形边界的排斥性与吸引性,特别地,给出了离散系统异宿环的稳定性准则.根据边界不动点局部动力学性态我们建立了系统间的等价关系,并利用指标和公式对Leslie/Gomer映射与Atkinson/Allen映射进行了完整的等价分类.它们各自都具有33个稳定的等价类,其中第1-18类的每一条轨道都趋于一个不动点,然而其余的15类系统具有相对复杂的动力学性态.特别地,我们重点研究了Neimark-Sacker分支,周期震荡,异宿环的存在性及其稳定性等.II.对对于具有相同极小扰动内禀增长率的非自治LV系系统,建建立了解的分解公式,描描述其其长期动力学性态.我们首先建立解的分解公式:对具有相同时间依赖的内禀增长率的LotkaVolterra(LV)系统的解可以表示为未受扰动的自治系统的解与一个一维的受相同扰动函数扰动的非自治Logistic方程的解的乘积.由此可知,扰扰动后的非自治LV系系统将继承自治LV系系统的全部动力学性态.利用分解公式,我们首次给出了周期/几乎周期扰动的竞争LV系统拥有拟周期解/几乎周期解及混动运动的存在性结果.基于Zeeman的nullcline分类结果,我们完整地分类出具有相同内禀增长率的3-维连续竞争LV模型的动力学性态的37种拓扑类型,进而利用分解公式,我们对受扰动后的非自治竞争LV模型生成的斜积流以拉回轨道的方式给出了对应的全局动力学性态的分类.III.证证明了Zeeman对3-维维竞争LV系系统的nullcline分分类适用于具有线性nullclines结结构的的三维竞争ODEs系系统.我们将开发Zeeman关于3-维竞争LV系统的nullcline分类在更广的三维竞争Kolmogorov系统中的适用性.证明了具有线性nullclines结构的三维竞争Kolmogorov系统按照nullcline等价关系总的分成33个稳定的nullcline等价类,其中前25类具有平凡动力学、第27类具有异缩环、第32类不可能发生Hopf分支.通过研究Hopf分支发现:3-维连续的竞争Ricker模型和Leslie/Gower模型在26-31类均能发生Hopf分支,然而,Atkinson/Allen模型和Gompertz模型仅在第26,27类中发生Hopf分支.我们还比较各系统动力学之间的区别.此外,我们还给出Ricker模型、Leslie/Gower模型和Gompertz模型存在两个极限环的例子.这一发现大大地推广了Zeeman的nullcline分类方法的应用.
[Abstract]:This paper mainly deals with the discrete dynamic systems induced by competitive mapping, the non autonomous system of time recovery and the long-term dynamic state of the autonomous competitive system. The main research work includes the following three aspects: I. establishes the existence of the existence of the existence of the load mono form for a general competition mapping mapping. The equivalence classification is given for the Leslie/Gower mapping and Atkinson/Allen mapping of three-dimensional dimension. First, we prove the existence and uniqueness of the existence and uniqueness of the load monomiform for a highly verifiable competition mapping. Therefore, it is proved that a large class of arbitrary dimensional discrete competition systems have load monomers. In particular, it is the first time that the arbitrary dimension Leslie/Gomer mapping and Atkinson/Allen mapping have unconditional existence of load monforms. Based on the theory of the load monomer existence, we give the index and formula on the 3- dimension competitive projection monomer and study the negative effect. In particular, the stability criterion of the discrete system is given, and the equivalence relation between the systems is established according to the local dynamic state of the fixed point of the boundary, and the equivalent classification of the Leslie/Gomer mapping and the Atkinson/Allen mapping is carried out by the index and the formula. There are 33 stable equivalence classes, of which each of the 1-18 classes tends to a fixed point, but the rest of the 15 systems have relatively complex dynamic states. In particular, we focus on the Neimark-Sacker bifurcation, periodic oscillations, the existence of the heteroclinic rings and the stability of the.II. pairs with the same minimum perturbations. The long rate nonautonomous LV system is built to establish the decomposition formula of the understanding and describe its long-term dynamic state. First, we establish the decomposition formula of the solution: the solution of the LotkaVolterra (LV) system with the intrinsic growth rate of the same time dependence can be expressed as the solution of the undisturbed self governing system and a one dimension of the same perturbation function. The product of the solution of the perturbed non autonomous Logistic equation shows that the nonautonomous LV system system after scrambling will inherit all the dynamic states of the autonomous LV system system. By using the decomposition formula, we first give the existence results of the quasi periodic solution / several periodic solutions and the mixed motion of the competitive LV system with periodic / almost periodic perturbation. Based on the Zeeman nullcline classification results, we completely classify the 37 topological types of the dynamic state of the 3- dimensional continuous competitive LV model with the same intrinsic growth rate, and then use the decomposition formula, we give the corresponding global dynamics for the diagonal flow generated by the disturbed non autonomous competitive LV model in the way of pulling back the orbit. The classified.III. certificate proves that the Zeeman's nullcline classification for the 3- VD LV system system is suitable for the three-dimensional competitive ODEs system with a linear nullclines junction structure. We will develop Zeeman's applicability to the 3- dimension competitive LV system in the wider three-dimensional competition Kolmogorov system. The three dimensional competitive Kolmogorov system of the linear nullclines structure is divided into 33 stable nullcline equivalence classes according to the nullcline equivalence relation, of which the first 25 classes have ordinary dynamics, the twenty-seventh classes have the contraction rings and the thirty-second classes cannot have the Hopf branch. By studying the Hopf branch, the 3- dimension continuous competitive Ricker model and the Leslie/Gower module are found. The Hopf branch can occur in all 26-31 classes, however, the Atkinson/Allen model and the Gompertz model occur only in the Hopf branch in the 26,27 class. We also compare the differences between the system dynamics. In addition, we also give examples of the Ricker model, the Leslie/Gower model and the Gompertz model with two limit cycles. This discovery greatly promotes Z The application of Eeman's nullcline classification method.
【学位授予单位】：上海师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175
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本文编号：1987857