本文選題：競(jìng)爭(zhēng)動(dòng)力系統(tǒng) + 離散競(jìng)爭(zhēng)映射��； 參考：《上海師范大學(xué)》2016年博士論文

【摘要】：本文主要研究由競(jìng)爭(zhēng)映射誘導(dǎo)的離散動(dòng)力系統(tǒng)、時(shí)間回復(fù)的非自治系統(tǒng)和自治競(jìng)爭(zhēng)系統(tǒng)的長(zhǎng)期動(dòng)力學(xué)性態(tài),主要研究工作包括如下三方面:I.對(duì)對(duì)于一一般的競(jìng)競(jìng)爭(zhēng)映映射建建立了了負(fù)載載單形形的存存在性性理論論及指指標(biāo)公公式,并并由此根根據(jù)邊邊界不不動(dòng)點(diǎn)點(diǎn)的局局部動(dòng)動(dòng)力學(xué)學(xué)性態(tài)態(tài)定義義了等等價(jià)關(guān)關(guān)系,對(duì)對(duì)三維維的Leslie/Gower映映射及Atkinson/Allen映映射給出了等價(jià)分類(lèi).首先,我們證明了一個(gè)十分容易驗(yàn)證的競(jìng)爭(zhēng)映射的負(fù)載單形存在唯一性定理,由此證明了一大類(lèi)的任意維離散競(jìng)爭(zhēng)系統(tǒng)都存在負(fù)載單形,特別地,首次證明了任意維的Leslie/Gomer映射與Atkinson/Allen映射等無(wú)條件存在負(fù)載單形.基于負(fù)載單形存在性理論,我們給出了3-維競(jìng)爭(zhēng)映射負(fù)載單形上的指標(biāo)和公式并研究了負(fù)載單形邊界的排斥性與吸引性,特別地,給出了離散系統(tǒng)異宿環(huán)的穩(wěn)定性準(zhǔn)則.根據(jù)邊界不動(dòng)點(diǎn)局部動(dòng)力學(xué)性態(tài)我們建立了系統(tǒng)間的等價(jià)關(guān)系,并利用指標(biāo)和公式對(duì)Leslie/Gomer映射與Atkinson/Allen映射進(jìn)行了完整的等價(jià)分類(lèi).它們各自都具有33個(gè)穩(wěn)定的等價(jià)類(lèi),其中第1-18類(lèi)的每一條軌道都趨于一個(gè)不動(dòng)點(diǎn),然而其余的15類(lèi)系統(tǒng)具有相對(duì)復(fù)雜的動(dòng)力學(xué)性態(tài).特別地,我們重點(diǎn)研究了Neimark-Sacker分支,周期震蕩,異宿環(huán)的存在性及其穩(wěn)定性等.II.對(duì)對(duì)于具有相同極小擾動(dòng)內(nèi)稟增長(zhǎng)率的非自治LV系系統(tǒng),建建立了解的分解公式,描描述其其長(zhǎng)期動(dòng)力學(xué)性態(tài).我們首先建立解的分解公式:對(duì)具有相同時(shí)間依賴(lài)的內(nèi)稟增長(zhǎng)率的LotkaVolterra(LV)系統(tǒng)的解可以表示為未受擾動(dòng)的自治系統(tǒng)的解與一個(gè)一維的受相同擾動(dòng)函數(shù)擾動(dòng)的非自治Logistic方程的解的乘積.由此可知,擾擾動(dòng)后的非自治LV系系統(tǒng)將繼承自治LV系系統(tǒng)的全部動(dòng)力學(xué)性態(tài).利用分解公式,我們首次給出了周期/幾乎周期擾動(dòng)的競(jìng)爭(zhēng)LV系統(tǒng)擁有擬周期解/幾乎周期解及混動(dòng)運(yùn)動(dòng)的存在性結(jié)果.基于Zeeman的nullcline分類(lèi)結(jié)果,我們完整地分類(lèi)出具有相同內(nèi)稟增長(zhǎng)率的3-維連續(xù)競(jìng)爭(zhēng)LV模型的動(dòng)力學(xué)性態(tài)的37種拓?fù)漕?lèi)型,進(jìn)而利用分解公式,我們對(duì)受擾動(dòng)后的非自治競(jìng)爭(zhēng)LV模型生成的斜積流以拉回軌道的方式給出了對(duì)應(yīng)的全局動(dòng)力學(xué)性態(tài)的分類(lèi).III.證證明了Zeeman對(duì)3-維維競(jìng)爭(zhēng)LV系系統(tǒng)的nullcline分分類(lèi)適用于具有線性nullclines結(jié)結(jié)構(gòu)的的三維競(jìng)爭(zhēng)ODEs系系統(tǒng).我們將開(kāi)發(fā)Zeeman關(guān)于3-維競(jìng)爭(zhēng)LV系統(tǒng)的nullcline分類(lèi)在更廣的三維競(jìng)爭(zhēng)Kolmogorov系統(tǒng)中的適用性.證明了具有線性nullclines結(jié)構(gòu)的三維競(jìng)爭(zhēng)Kolmogorov系統(tǒng)按照nullcline等價(jià)關(guān)系總的分成33個(gè)穩(wěn)定的nullcline等價(jià)類(lèi),其中前25類(lèi)具有平凡動(dòng)力學(xué)、第27類(lèi)具有異縮環(huán)、第32類(lèi)不可能發(fā)生Hopf分支.通過(guò)研究Hopf分支發(fā)現(xiàn):3-維連續(xù)的競(jìng)爭(zhēng)Ricker模型和Leslie/Gower模型在26-31類(lèi)均能發(fā)生Hopf分支,然而,Atkinson/Allen模型和Gompertz模型僅在第26,27類(lèi)中發(fā)生Hopf分支.我們還比較各系統(tǒng)動(dòng)力學(xué)之間的區(qū)別.此外,我們還給出Ricker模型、Leslie/Gower模型和Gompertz模型存在兩個(gè)極限環(huán)的例子.這一發(fā)現(xiàn)大大地推廣了Zeeman的nullcline分類(lèi)方法的應(yīng)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.
【學(xué)位授予單位】：上海師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O175
，

本文編號(hào)：1987857