几类高阶非线性差分方程的稳定性及应用
发布时间：2019-05-29 06:18
【摘要】：差分方程（系统）是描述现实世界中随离散时间演化规律的有力建模工具，自然界和人类社会中的很多现象都可以用适当的差分方程模型来刻画。例如，差分方程在算法分析、种群动力学、经济学等领域都有着广泛的应用。不仅如此，许多连续的数学模型也可以通过“离散化”转换为相应的离散形式，进而可以用计算机进行数值模拟。近年来，高阶差分方程、非自治差分方程、Max型差分方程和差分系统是差分方程领域的研究热点。 本学位论文主要研究了几类高阶差分方程（系统）的全局渐近稳定性和全局吸引性，并通过差分方程建模建立了一个含有用户意识度的网络蠕虫传播模型。具体来说，取得的主要研究成果如下： ①研究了两类含有实指数的有理型差分方程。首先，利用变换法研究了一类高阶Stevi方程及其对应的Max型方程，在一定条件下得到了解的渐近表达式。其次，运用部分度量（Part metric）方法研究了一类含有指数参数的高阶对称有理差分方程。通过建立与部分度量有关的不等式链，证明了如果所有指数参数的绝对值都小于或等于1，则方程的唯一正平衡点是全局渐近稳定的。 ②研究了两类含有抽象函数的差分方程的全局吸引性。首先，研究了一类含有抽象函数的非自治差分方程。通过构造交错序列，证明了在一定参数条件下方程的平衡点是全局吸引子。其次，首次提出了基于此非自治方程的一类Max型自治方程，并研究了它的一种特殊情况。采用类似的方法，证明了在一定参数条件下Max型方程的解具有全局吸引性。 ③研究了一类非自治Max型差分方程的全局吸引性。首先，研究了含有单一非自治项的情况，在几组不同的非自治项及参数满足条件下分别证明了其解具有全局吸引性。随后，对方程中含有多个非自治项的情况进行了研究，同样给出了几组方程中各参数满足的充分条件，，使得其解具有全局吸引性。 ④研究了两类差分系统的动力学性质。首先，研究了一类一般的二维差分系统的全局吸引性。具体地说，在一定条件下证明了系统的唯一正平衡点是全局吸引子。其次，研究了一类高阶循环差分系统。通过定义矩阵上的部分度量，证明了其平衡点的全局稳定性。 ⑤通过差分方程建模建立了一个含有用户意识度的网络蠕虫传播模型—dSLB模型。运用差分方程稳定性理论对模型的动力学性质进行了分析，得到了影响蠕虫传播动力学行为的阈值参数。具体地说，分析了模型在无蠕虫平衡点及蠕虫平衡点处的动力学性质，在一定条件下从理论上证明了当阈值参数小于1时无蠕虫平衡点是渐近稳定的，而当阈值参数大于1时蠕虫平衡点是渐近稳定的。
[Abstract]:Difference equation (system) is a powerful modeling tool to describe the evolution of discrete time in the real world. Many phenomena in nature and human society can be characterized by appropriate difference equation model. For example, difference equations are widely used in algorithm analysis, population dynamics, economics and other fields. Not only that, many continuous mathematical models can also be transformed into corresponding discrete forms by "discretization", and then numerical simulation can be carried out by computer. In recent years, higher order difference equation, nonautonomous difference equation, Maxtype difference equation and difference system are the research hotspots in the field of difference equation. In this thesis, we mainly study the global asymptotic stability and global attractiveness of several kinds of high-order difference equations (systems), and establish a network worm propagation model with user awareness through difference equation modeling. Specifically, the main research results are as follows: (1) two kinds of rational difference equations with real indices are studied. Firstly, a class of higher order Stevi equations and their corresponding Maxtype equations are studied by using the transformation method, and the asymptotic expressions of the solutions are obtained under certain conditions. Secondly, a class of higher order symmetric rational difference equations with exponential parameters is studied by using the partial metric (Part metric) method. By establishing the inequality chain related to partial metrics, it is proved that if the absolute values of all exponential parameters are less than or equal to 1, then the only positive equilibrium point of the equation is globally asymptotically stable. (2) the global attractivity of two kinds of difference equations with abstract functions is studied. Firstly, a class of nonautonomous difference equations with abstract functions is studied. By constructing the staggered sequence, it is proved that the equilibrium point of the equation is a global Attractor under certain parameter conditions. Secondly, a class of Maxtype autonomous equations based on this nonautonomous equation is proposed for the first time, and a special case of Maxtype autonomous equations is studied. By using a similar method, it is proved that the solution of Max equation has global attraction under certain parameters. (3) the global attractivity of a class of nonautonomous Maxtype difference equations is studied. Firstly, the case with a single nonautonomous term is studied, and it is proved that the solution has global attractiveness under several different groups of nonautonomous terms and under the condition that the parameters satisfy the conditions. Then, the case of multiple nonautonomous terms in the equation is studied, and the sufficient conditions for the parameters in several sets of equations to be satisfied are also given, so that the solution has global attractiveness. (4) the dynamic properties of two kinds of difference systems are studied. Firstly, the global attractiveness of a general class of two-dimensional difference systems is studied. Specifically, under certain conditions, it is proved that the only positive equilibrium point of the system is the global Attractor. Secondly, a class of high-order cyclic difference systems is studied. By defining some measures on the matrix, the global stability of the equilibrium point is proved. (5) A network worm propagation model with user consciousness, dSLB model, is established by difference equation modeling. The dynamic properties of the model are analyzed by using the stability theory of difference equation, and the threshold parameters affecting the dynamic behavior of worm propagation are obtained. Specifically, the dynamic properties of the model at worm-free equilibrium point and worm-free equilibrium point are analyzed. Under certain conditions, it is proved theoretically that the worm-free equilibrium point is asymptotically stable when the threshold parameter is less than 1. When the threshold parameter is greater than 1, the worm equilibrium point is asymptotically stable.
【学位授予单位】：重庆大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：O175.7;TP393.08
本文编号：2487720
[Abstract]:Difference equation (system) is a powerful modeling tool to describe the evolution of discrete time in the real world. Many phenomena in nature and human society can be characterized by appropriate difference equation model. For example, difference equations are widely used in algorithm analysis, population dynamics, economics and other fields. Not only that, many continuous mathematical models can also be transformed into corresponding discrete forms by "discretization", and then numerical simulation can be carried out by computer. In recent years, higher order difference equation, nonautonomous difference equation, Maxtype difference equation and difference system are the research hotspots in the field of difference equation. In this thesis, we mainly study the global asymptotic stability and global attractiveness of several kinds of high-order difference equations (systems), and establish a network worm propagation model with user awareness through difference equation modeling. Specifically, the main research results are as follows: (1) two kinds of rational difference equations with real indices are studied. Firstly, a class of higher order Stevi equations and their corresponding Maxtype equations are studied by using the transformation method, and the asymptotic expressions of the solutions are obtained under certain conditions. Secondly, a class of higher order symmetric rational difference equations with exponential parameters is studied by using the partial metric (Part metric) method. By establishing the inequality chain related to partial metrics, it is proved that if the absolute values of all exponential parameters are less than or equal to 1, then the only positive equilibrium point of the equation is globally asymptotically stable. (2) the global attractivity of two kinds of difference equations with abstract functions is studied. Firstly, a class of nonautonomous difference equations with abstract functions is studied. By constructing the staggered sequence, it is proved that the equilibrium point of the equation is a global Attractor under certain parameter conditions. Secondly, a class of Maxtype autonomous equations based on this nonautonomous equation is proposed for the first time, and a special case of Maxtype autonomous equations is studied. By using a similar method, it is proved that the solution of Max equation has global attraction under certain parameters. (3) the global attractivity of a class of nonautonomous Maxtype difference equations is studied. Firstly, the case with a single nonautonomous term is studied, and it is proved that the solution has global attractiveness under several different groups of nonautonomous terms and under the condition that the parameters satisfy the conditions. Then, the case of multiple nonautonomous terms in the equation is studied, and the sufficient conditions for the parameters in several sets of equations to be satisfied are also given, so that the solution has global attractiveness. (4) the dynamic properties of two kinds of difference systems are studied. Firstly, the global attractiveness of a general class of two-dimensional difference systems is studied. Specifically, under certain conditions, it is proved that the only positive equilibrium point of the system is the global Attractor. Secondly, a class of high-order cyclic difference systems is studied. By defining some measures on the matrix, the global stability of the equilibrium point is proved. (5) A network worm propagation model with user consciousness, dSLB model, is established by difference equation modeling. The dynamic properties of the model are analyzed by using the stability theory of difference equation, and the threshold parameters affecting the dynamic behavior of worm propagation are obtained. Specifically, the dynamic properties of the model at worm-free equilibrium point and worm-free equilibrium point are analyzed. Under certain conditions, it is proved theoretically that the worm-free equilibrium point is asymptotically stable when the threshold parameter is less than 1. When the threshold parameter is greater than 1, the worm equilibrium point is asymptotically stable.
【学位授予单位】：重庆大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：O175.7;TP393.08
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