Research On Three Kinds Of Second Order Rational Difference Equations

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=?xn/(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.