【摘要】：差分方程（系統(tǒng)）是描述現實世界中隨離散時間演化規(guī)律的有力建模工具，自然界和人類社會中的很多現象都可以用適當的差分方程模型來刻畫。例如，差分方程在算法分析、種群動力學、經濟學等領域都有著廣泛的應用。不僅如此，許多連續(xù)的數學模型也可以通過“離散化”轉換為相應的離散形式，進而可以用計算機進行數值模擬。近年來，高階差分方程、非自治差分方程、Max型差分方程和差分系統(tǒng)是差分方程領域的研究熱點。 本學位論文主要研究了幾類高階差分方程（系統(tǒng)）的全局漸近穩(wěn)定性和全局吸引性，并通過差分方程建模建立了一個含有用戶意識度的網絡蠕蟲傳播模型。具體來說，取得的主要研究成果如下： ①研究了兩類含有實指數的有理型差分方程。首先，利用變換法研究了一類高階Stevi方程及其對應的Max型方程，在一定條件下得到了解的漸近表達式。其次，運用部分度量（Part metric）方法研究了一類含有指數參數的高階對稱有理差分方程。通過建立與部分度量有關的不等式鏈，證明了如果所有指數參數的絕對值都小于或等于1，則方程的唯一正平衡點是全局漸近穩(wěn)定的。 ②研究了兩類含有抽象函數的差分方程的全局吸引性。首先，研究了一類含有抽象函數的非自治差分方程。通過構造交錯序列，證明了在一定參數條件下方程的平衡點是全局吸引子。其次，首次提出了基于此非自治方程的一類Max型自治方程，并研究了它的一種特殊情況。采用類似的方法，證明了在一定參數條件下Max型方程的解具有全局吸引性。 ③研究了一類非自治Max型差分方程的全局吸引性。首先，研究了含有單一非自治項的情況，在幾組不同的非自治項及參數滿足條件下分別證明了其解具有全局吸引性。隨后，對方程中含有多個非自治項的情況進行了研究，同樣給出了幾組方程中各參數滿足的充分條件，，使得其解具有全局吸引性。 ④研究了兩類差分系統(tǒng)的動力學性質。首先，研究了一類一般的二維差分系統(tǒng)的全局吸引性。具體地說，在一定條件下證明了系統(tǒng)的唯一正平衡點是全局吸引子。其次，研究了一類高階循環(huán)差分系統(tǒng)。通過定義矩陣上的部分度量，證明了其平衡點的全局穩(wěn)定性。 ⑤通過差分方程建模建立了一個含有用戶意識度的網絡蠕蟲傳播模型—dSLB模型。運用差分方程穩(wěn)定性理論對模型的動力學性質進行了分析，得到了影響蠕蟲傳播動力學行為的閾值參數。具體地說，分析了模型在無蠕蟲平衡點及蠕蟲平衡點處的動力學性質，在一定條件下從理論上證明了當閾值參數小于1時無蠕蟲平衡點是漸近穩(wěn)定的，而當閾值參數大于1時蠕蟲平衡點是漸近穩(wěn)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.
【學位授予單位】：重慶大學
【學位級別】：博士
【學位授予年份】：2014
【分類號】：O175.7;TP393.08

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