本文選題：右端不連續(xù)微分方程 + Σ-奇異點(diǎn)��； 參考：《湖南大學(xué)》2015年博士論文

【摘要】：由于數(shù)學(xué)分析工具的缺乏,右端不連續(xù)微分方程理論的發(fā)展相當(dāng)緩慢.至今,右端不連續(xù)微分方程理論的發(fā)展仍處于初級階段,很多理論都沒有得到完善.為此,本學(xué)位論文首先發(fā)展了有關(guān)右端不連續(xù)非自治廣義齊次微分方程的穩(wěn)定性理論,主要討論了右端不連續(xù)非自治廣義齊次微分方程以及一類具有不連續(xù)擾動(dòng)項(xiàng)的齊次系統(tǒng)解的收斂方式.然后,我們根據(jù)現(xiàn)實(shí)生活中一些食餌和捕食者之間的接觸特征,建立了一個(gè)具有不連續(xù)功能反應(yīng)函數(shù)的食餌捕食模型,并應(yīng)用右端不連續(xù)微分方程定性理論對該模型進(jìn)行了深入的研究.在本學(xué)位論文的最后兩部分,我們通過構(gòu)造Lyapunov函數(shù)的方法分別討論了一類光滑的HIV雙倉室模型和一類具有不連續(xù)激勵(lì)函數(shù)的神經(jīng)網(wǎng)絡(luò)模型平衡態(tài)的穩(wěn)定性.值得一提的是,在研究具有不連續(xù)激勵(lì)函數(shù)的神經(jīng)網(wǎng)絡(luò)模型平衡態(tài)的穩(wěn)定性時(shí),我們獲得了在一定的條件下,平衡態(tài)是有限時(shí)間收斂的.該收斂方式在光滑的HIV倉室模型中是不可能具有的.具體地說,本學(xué)位論文主要分為六章. 在第一章中,我們首先回顧了右端不連續(xù)微分方程和穩(wěn)定性理論的研究歷史和發(fā)展概況.隨后,我們逐一地介紹了不連續(xù)生物動(dòng)力學(xué)、HIV動(dòng)力學(xué)和不連續(xù)神經(jīng)網(wǎng)絡(luò)動(dòng)力學(xué)的研究歷史和發(fā)展現(xiàn)狀. 在第二章中,我們簡要地介紹了本學(xué)位論文所需的一些數(shù)學(xué)理論知識,主要包括Filippov系統(tǒng)的定性理論,如解的存在唯一性、解對初值和右端函數(shù)的依賴性、平面Filippov系統(tǒng)中不連續(xù)曲面上解的特性、∑-奇異點(diǎn)附近解的拓?fù)浣Y(jié)構(gòu).同時(shí), 一些有關(guān)光滑動(dòng)力系統(tǒng)和Filippov系統(tǒng)的穩(wěn)定性理論也在此處被列出. 在第三章中,我們完善了右端不連續(xù)非自治廣義齊次微分方程的穩(wěn)定性理論.利用齊次微分方程的特性(收縮解仍是原方程的解)和一些運(yùn)算上的處理技巧,得到了右端不連續(xù)非自治廣義齊次微分方程解的收斂特征：齊次度為零和正的非自治廣義齊次系統(tǒng),全局漸近穩(wěn)定的平衡點(diǎn)分別是指數(shù)穩(wěn)定和1/t型穩(wěn)定的.同時(shí),我們分析了一類具有內(nèi)外擾動(dòng)項(xiàng)的擾動(dòng)系統(tǒng)的動(dòng)力學(xué)行為,并將其與原系統(tǒng)進(jìn)行了對比研究.結(jié)果顯示該擾動(dòng)系統(tǒng)與原系統(tǒng)具有類似的動(dòng)力學(xué)行為,當(dāng)擾動(dòng)充分小時(shí),解的收斂方式在該擾動(dòng)下是穩(wěn)定的. 在第四章中,我們根據(jù)在現(xiàn)實(shí)生活中一些具有比率依賴的食餌與捕食者之間的接觸特性,建立了一個(gè)比率依賴的Filippov食餌捕食模型.主要運(yùn)用右端不連續(xù)微分方程定性理論中有關(guān)∑-奇異點(diǎn)拓?fù)浣Y(jié)構(gòu)的相關(guān)知識,對該模型所有∑-奇異點(diǎn)的局部結(jié)構(gòu)進(jìn)行了詳細(xì)的分析.此外,對于不同的參數(shù),我們得到了該模型具有14種不同的全局動(dòng)力學(xué)行為.特別地,對于一定范圍內(nèi)的參數(shù),全局穩(wěn)定的偽平衡點(diǎn)和全局有限時(shí)間穩(wěn)定的周期解是存在的,并且所得的部分動(dòng)力學(xué)行為能合理地解釋部分實(shí)驗(yàn)和現(xiàn)實(shí)中的現(xiàn)象. 在第五章中,我們將構(gòu)造Lyapunov函數(shù)的方法與LaSalle不變原理相結(jié)合,分別討論了一類光滑的七維和八維HIV倉室模型平衡態(tài)的漸近穩(wěn)定性.通過對這兩個(gè)模型的分析,我們得到了:當(dāng)基本再生數(shù)小于等于1時(shí)(即感染平衡點(diǎn)不存在),無病平衡點(diǎn)是全局漸近穩(wěn)定的;當(dāng)基本再生數(shù)大于1時(shí)(即感染平衡點(diǎn)存在),感染平衡點(diǎn)是全局漸近穩(wěn)定的. 在第六章中,根據(jù)周期解是否與不連續(xù)曲面相切,我們研究了一類具有不連續(xù)激勵(lì)函數(shù)的Hopfeld神經(jīng)網(wǎng)絡(luò)模型周期解的有限時(shí)間收斂性.當(dāng)周期解和不連續(xù)曲面相切時(shí),周期解一定是有限時(shí)間穩(wěn)定的.當(dāng)周期解和所有不連續(xù)曲面橫截相交時(shí),在一定的條件下,通過構(gòu)造Lyapunov函數(shù)法可得周期解的有限時(shí)間收斂性.
[Abstract]:Due to the lack of tools of mathematical analysis, discontinuous development theory of differential equations is very slow. So far, the right end is not continuous development of the theory of differential equation is still in the primary stage, many theories are not perfect. Therefore, this thesis firstly developed the discontinuous nonautonomous generalized homogeneous differential equation stability the theory, mainly discusses the discontinuous nonautonomous generalized homogeneous differential equations and a class of discontinuous solutions of the perturbation convergence mode of homogeneous system. Then, we according to the contact characteristics between the real life of some prey and predator, has not established a continuous functional response predator-prey model and application of discontinuous qualitative theory of differential equations of the model are studied. In the last two parts of this thesis, we construct the Lyapunov function. Method are used to discuss the stability of a class of smooth HIV double compartment model and a neural network model of equilibrium discontinuous activation function. It is worth mentioning that the stability of neural networks model with equilibrium discontinuous activation functions in the research, we obtained under certain conditions, the equilibrium state finite time convergence. The convergence is not possible with the HIV chamber smooth model. Specifically, this thesis is mainly divided into six chapters.
In the first chapter, we first review the research history and development of the right end discontinuous differential equations and stability theory. Next, we introduce the history and development of discontinuous biologic dynamics, HIV dynamics and discontinuous neural network dynamics.
In the second chapter, we briefly introduce some mathematical theory in this thesis is required, including the qualitative theory of Filippov system, such as the existence and uniqueness of solutions, solutions to the initial value and the right function dependent characteristics of planar Filippov system of discontinuous surface solution, solution of the topological structure near sigma singular points. At the same time,
Some stability theories about smooth dynamic systems and Filippov systems are also listed here.
In the third chapter, we improve the discontinuous nonautonomous generalized homogeneous differential equation stability theory. Based on the characteristics of homogeneous differential equations (contraction of solution is the solution of the equation) and some computing skills, the discontinuous non convergence of generalized homogeneous differential equations autonomy: homogeneous zero and positive non autonomous generalized homogeneous system, global asymptotic stability of the equilibrium point are exponential stability and 1/t stability. At the same time, we analyze the dynamic behavior of a class of perturbed system of internal and external disturbances, which were compared with the original system. The results showed the disturbance of system and the original system has similar dynamical behavior, when the perturbation is sufficiently small, the convergence solutions is stable in the disturbance.
In the fourth chapter, according to the US in real life has some contact characteristics between prey and predator ratio dependent, established Filippov predator-prey model with a ratio dependent. Mainly using the discontinuous of sigma in qualitative theory of differential equation of singular point related knowledge of topology, the local structure of the the model of all sigma singular points are analyzed in detail. In addition, for different parameters, we get the model with 14 different global dynamics. In particular, the parameters for a certain range, the global stability of the equilibrium point and periodic pseudo global finite time stable solutions exist, dynamics the behavior and can reasonably explain some experiments and the reality of the phenomenon.
In the fifth chapter, we will construct the Lyapunov function method and LaSalle invariance principle combined respectively asymptotic stability of a class of smooth seven and eight dimensional HIV compartment model of equilibrium is discussed. Through the analysis of these two models, we obtained: when the basic reproduction number is smaller than or equal to 1 (i.e. the infected equilibrium does not exist), the disease-free equilibrium is globally asymptotically stable; when the basic reproduction number is greater than 1 (i.e. the infection equilibrium exists), the infection free equilibrium is globally asymptotically stable.
In the sixth chapter, according to the periodic solution and discontinuous tangent to the surface, we study a class of finite time convergence of periodic Hopfeld neural network model of discontinuous solution. When the excitation function of periodic solutions and discontinuous surface tangent, periodic solutions are finite time stable. When the periodic solutions and all discontinuities surface transverse intersection, under certain conditions, by constructing Lyapunov function method can converge in finite time of periodic solutions.

【學(xué)位授予單位】：湖南大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2015
【分類號】：O175

【參考文獻(xiàn)】

相關(guān)博士學(xué)位論文 前3條

1 郭振遠(yuǎn);右端不連續(xù)微分方程理論及其應(yīng)用[D];湖南大學(xué);2009年

2 李立平;具有連續(xù)或不連續(xù)輸出函數(shù)的神經(jīng)網(wǎng)絡(luò)模型的動(dòng)力學(xué)研究[D];湖南大學(xué);2009年

3 王佳伏;時(shí)滯微分包含初值問題與穩(wěn)定性的理論及應(yīng)用[D];湖南大學(xué);2009年

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本文編號：1741119