【作者】 王媛媛；

【导师】 黄庆道；

【作者基本信息】 吉林大学 ， 运筹学与控制论， 2019， 硕士

【摘要】 差分方程是一种离散的数学模型,主要展现的是一种变化规律,而它的应用由来是源于现实生活中的一些问题,这些问题无法用微分方程或连续的函数进行计算解析,所以就需要将其离散化,而差分方程在离散方程中占据着重要地位,对于上述情况就突显了其优势.另外,随着近几年信息和科技的不断进步,差分方程的理论和方法逐渐完善,并且通过计算机对差分方程迭代模拟时速度变快,可以更有效的处理实际问题,而且可以直观地看出研究对象的变化规律,所以在各行各业的应用也越来越广泛,比如在经济学、生物学、医学等领域具有着重要意义.本文主要研究的是参数和初始值都是正实数的三类二阶有理差分方程的稳定性、吸引性、收敛性及周期性的相关问题,并给出了相应的结论及内容.本文共分为五章:第一章主要是介绍了有理差分方程的研究历史、背景及现状.第二章先是介绍了线性稳定性理论、“M&m”方法、庞加莱定理等本文所需的理论知识,然后讨论了如下的二阶差分方程xn+1=pxn+qxn-1/xn+xn-1,n=0,1,…,(0,1)其中参数和初始值皆为正实数.通过研究我们知道了方程(0.1)的平衡解是全局渐近稳定的,并得到了平衡解收敛的结论,最后通过数值分析作出图像验证了结论的正确性.第三章主要是对如下方程进行了研究xn+1=xn+pxn-1/qxn,n=0,1,…,(0,2)其中参数和初始值皆为正实数.在这章中根据线性稳定性的定理及“M&m”方法得到了当p<1时,方程(0.2)的平衡解是全局渐近稳定的,当p>1时,方程(0.2)的平衡解是不稳定的,而p=1时,方程(0.2)存在一个正2-周期解,并通过稳定子空间定理及线性稳定性定理证明了此2-周期解是不稳定的,最后通过Matlab数值分析作出图像验证了结论的正确性.第四章主要是对如下方程进行了研究xn+1=axn/(1+xn)xn-1，n=0,1…，这里的参数和初始值也为正实数.在这章中根据线性稳定性定理得到了方程的平衡解是不稳定的,再由“M&m”方法证明了其平衡解是一个全局吸引子,接下来通过研究发现了方程(0.3)存在一个正5-周期解,并且其周期解随初始值改变而变化,最后通过Matlab数值分析作出图像验证了结论的正确性.第五章主要是对本文进行了全方面的总结并给出了一些期望.更多还原

【Abstract】 The difference equation is a discrete mathematical model,which mainly shows a change law,and its application originates from some problems in real life.These problems cannot be solved by differential equations or continuous functions,so It needs to be discretized,and the difference equation occupies an important position in the discrete equation,which highlights its advantages.In addition,with the continuous advancement of information and technolo-gy in recent years,the theory and method of the difference equation have been gradually improved,and the speed of the iterative simulation of the difference equation is faster by computer,which can effectively deal with practical problems,and can be visually seen.The changing laws of the research objects are more and more widely used in various industries,such as economics,biology,medicine and other fields.In this paper,the stability,attractivi-ty,convergence and periodicity of three kinds of second-order rational difference equations with positive parameters and initial values are studied.The corresponding conclusions and contents are given.The paper is divided into five parts.chapter:The first chapter mainly introduces the research history,background and current situa-tion of rational difference equations.The second chapter introduces the linear stability theorem,"M&m" method,Poincaretheorem,etc.,and then discusses the following second-order difference equations.Xn+1=Pxn+qxn-1/xn+xn-1,n=0,1…(0.4)The parameters and initial values are all positive real numbers.Through the research we know that the equilibrium solution of the equation(0.4)is globally asymptotically stable,and the conclusion that the equilibrium solution converges is obtained.Finally,the numerical analysis is made.The image verifies the accuracy of the conclusion.The third chapter mainly studies the following equations.xn+1=xn+pxn-1/qxn,n=0,1…(0.5)The parameters and initial values are positive real numbers.In this chapter,according to the linear stability theorem and the "M&m" method,whenp<1,the equilibrium solution of equation(0.5)is globally asymptotically stable.When p>1,the equilibrium solution of the equation(0.5)is unstable,and when p=1,the equation(0.5)has a positive 2-periodic solution,and it is proved that the 2-periodic solution is unstable by the stable subspace theorem and the linear stability theorem.Finally,image verification is performed by Matlab numerical analysis.The accuracy of the conclusion.The fourth chapter mainly studies the following equations.xn+1=αxn/(1+xn)xn-1,n=0,1,…(0.6)The parameters and initial values here are also positive real numbers.In this chapter,accord-ing to the linear stability theorem,the equation is unstable,and then the "M&m" method proves that the equilibrium solution is a global attractor.Then we found that the equation(0.6)has a positive 5-period solution,and its periodic solution changes with the initial val-ue.Finally,the correctness of the conclusion is verified by Matlab numerical analysis.The fifth chapter is mainly to summarize the whole article and give some expectations.更多还原