本文選題：隨機動力系統(tǒng)　切入點：經(jīng)濟模型　出處：《廣西師范學(xué)院》2016年碩士論文　論文類型：學(xué)位論文

【摘要】：隨著自然科學(xué)的不斷發(fā)展,人們對現(xiàn)實世界的認(rèn)識越來越貼近本質(zhì).因此,現(xiàn)實系統(tǒng)中不可避免的隨機和非線性因素已成為眾多數(shù)學(xué)家和其它領(lǐng)域科學(xué)家關(guān)注的焦點.特別是,近年來經(jīng)濟金融、生物系統(tǒng)等諸多領(lǐng)域中已推導(dǎo)出大量的非線性隨機模型,進一步促使了人們對非線性隨機微分動力系統(tǒng)的理論和應(yīng)用的深入研究.為此,本文以隨機分析和隨機動力系統(tǒng)理論為工具,研究隨機綜合國力模型和一類非線性隨機金融混沌系統(tǒng)的動力學(xué)性質(zhì).具體研究內(nèi)容如下：第一章詳細陳述了研究問題的背景及意義以及本文所要用到的一些預(yù)備知識,同時介紹了本學(xué)位論文研究的主要內(nèi)容及其框架結(jié)構(gòu).第二章主要研究了隨機綜合國力模型的動力學(xué)性質(zhì).首先運用Poincare緊化變換將確定性綜合國力模型的無窮遠點轉(zhuǎn)化為有限坐標(biāo)點.進而,利用微分方程定性理論,研究其無窮遠點的動力學(xué)行為,包括結(jié)點和鞍點的穩(wěn)定性問題.其次,利用Lyapunov函數(shù),鞅不等式和隨機分析等方法和技巧,研究了Poincare緊化的隨機綜合國力模型的無窮遠點的動力學(xué)性質(zhì),包括全局解的存在性,漸近矩估計,解的穩(wěn)定性,矩均值,隨機最終有界性.最后,研究了隨機綜合國力模型的軌道估計.第三章主要研究了一類非線性隨機金融混沌系統(tǒng)的長期性行為.首先,利用截斷函數(shù)方法和隨機分析技巧,證明系統(tǒng)解的存在唯一性和有界性.其次,運用測度理論和Krylovs與Bogolyubov方法,證明系統(tǒng)存在平穩(wěn)分布.最后,利用隨機動力系統(tǒng)相關(guān)知識,證明系統(tǒng)存在唯一的隨機吸引子,并通過數(shù)值模擬進行了驗證.
[Abstract]:With the continuous development of natural science, people's understanding of the real world is more and more close to the essence. Therefore, the inevitable random and nonlinear factors in the real system have become the focus of attention of many mathematicians and scientists in other fields. In recent years, a large number of nonlinear stochastic models have been derived in many fields, such as economics, finance, biological systems, etc., which further promote the deep research on the theory and application of nonlinear stochastic differential dynamical systems. In this paper, stochastic analysis and stochastic dynamic system theory are used as tools. The dynamic properties of the stochastic comprehensive national strength model and a class of nonlinear stochastic financial chaotic systems are studied. The main contents are as follows: in chapter 1, the background and significance of the research and some preliminary knowledge to be used in this paper are described in detail. At the same time, the main contents and frame structure of this dissertation are introduced. In chapter 2, the dynamic properties of stochastic comprehensive national strength model are studied. Firstly, the infinity of deterministic comprehensive national strength model is transformed by Poincare compactness transformation. The point is transformed into a finite coordinate point. By using the qualitative theory of differential equations, the dynamical behavior of infinite points, including the stability of nodes and saddle points, is studied. Secondly, the methods and techniques of Lyapunov function, martingale inequality and stochastic analysis are used. In this paper, we study the dynamical properties of infinite points in Poincare compact stochastic comprehensive national strength model, including the existence of global solutions, asymptotic moment estimates, stability of solutions, mean moments, stochastic ultimate boundedness. In chapter 3, the long-term behavior of a class of nonlinear stochastic financial chaotic systems is studied. Firstly, the truncation function method and stochastic analysis technique are used. The existence, uniqueness and boundedness of the solution of the system are proved. Secondly, the existence of stationary distribution of the system is proved by means of measure theory and Krylovs and Bogolyubov methods. Finally, the existence of a unique random attractor is proved by using the relevant knowledge of the stochastic dynamical system. It is verified by numerical simulation.
【學(xué)位授予單位】：廣西師范學(xué)院
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2016
【分類號】：F224

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