两类分数阶数学模型的渐近稳定性与渐近周期性的研究

发布时间：2018-06-23 06:25

本文选题：分数阶 + Mittag-Leffler函数　；参考：《昆明理工大学》2017年硕士论文

【摘要】：本文主要应用Mittag-Leffler函数和分数阶微分方程的比较定理研究了两类分数阶数学模型的持久性、渐近稳定性与渐近周期性,推广并改进了已知文献中的结果.全文共分为五章.第一章主要阐述了选题的背景和意义,介绍了分数阶微分方程的发展、目前在各个领域的应用以及前人所做的相关工作.简要介绍了两类数学模型:食饵模型、Mackey-Glass呼吸道模型,而且对本文的主要工作做了简单介绍.第二章主要给出了分数阶微分方程的一些基本定义、引理和Mittag-Leffler函数的相关性质,并在这些性质的基础上,得到了一些重要的引理.为了得到模型的持久性,证明了分数阶微分方程的比较定理.为了研究模型的渐近稳定性及渐近周期性,引入了分数阶微分方程的特征方程.第三、四章通过运用Mittag-Leffler函数的性质、分数阶微分方程的比较定理以及拉普拉斯变换的特征方程与稳定性的关系,得到了分数阶食饵模型以及分数阶Mackey-Glass呼吸道模型的持久性、渐近稳定性与渐近周期性.最后,举例说明了我们研究结果的可行性.第五章主要对全文作一个简短的总结和回顾,并且探讨了未来的一些研究方向.
[Abstract]:In this paper, by using the comparison theorem of Mittag-Leffler function and fractional differential equation, we study the persistence, asymptotic stability and asymptotic periodicity of two kinds of fractional order mathematical models, and generalize and improve the known results. The full text is divided into five chapters. The first chapter describes the background and significance of the topic, introduces the development of fractional differential equations, the current applications in various fields and related work done by predecessors. This paper briefly introduces two kinds of mathematical models: the prey model and the Mackey-Glass respiratory tract model, and gives a brief introduction to the main work of this paper. In the second chapter, we give some basic definitions of fractional differential equations, the properties of Lemma and Mittag-Leffler function, and get some important Lemma on the basis of these properties. In order to obtain the permanence of the model, the comparison theorem of fractional differential equations is proved. In order to study the asymptotic stability and asymptotic periodicity of the model, the characteristic equations of fractional differential equations are introduced. In the third and fourth chapter, by using the properties of Mittag-Leffler function, the comparison theorem of fractional differential equation and the relation between the characteristic equation of Laplace transform and the stability, the persistence of fractional prey model and fractional Mackey-Glass respiratory model is obtained. Asymptotic stability and asymptotic periodicity. Finally, examples are given to illustrate the feasibility of our research results. Chapter five gives a brief summary and review of the full text, and discusses some research directions in the future.
【学位授予单位】：昆明理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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