Stability Analysis Of Rational Difference Equations Based On Iteration Method

Difference equation is a powerful tool describing laws of natural science and social science. Due to its great value and good prospects in various fields, theoretical research in difference equations has attracted many scholars. In recent years, rational difference equations and max-type difference equations have become a hot spot. Summarizing the existing related work, this thesis investigates dynamical behavior of several rational difference equations by the stability theory, convergence theorem, iterative method and mathematical induction method. The content mainly includes the following four parts:In the first part, dynamical behavior of a class of four-order rational difference equations is studied. By using linear theory and mathematical induction, the stability and boundedness of difference equations solutions are studied at equilibrium point. Additionally, the form of analytical solutions of the rational difference equation is obtained.In the second part, on the basis of the study on traditional stability of rational difference equations in the first part, global stability and asymptotic stability of a class of four-order nonlinear difference equations are studied. Through analyzing stability and convergence of the solution by the iteration method, some sufficient conditions to ensure global stability and asymptotical stability are obtained at the equilibrium point.In the third part, dynamical behavior of a class of rational max-type difference equations with exponent parameters is studied. By using inequalities techniques and logarithm to descend the exponent, the equation is simplified. Some sufficient conditions to ensure the boundedness of the solutions are obtained by iterative method and sequence analysis method and the convergence of the solutions with different parameters is verified.In the fourth part, periodicity of a class of max-type rational difference equation is studied. The periodicity of the low-order difference equations is acquired by iterative methods. While taking advantage of the linear theory, low-order equations are promoted to higher order equations and the periodicity of higher order difference equation with different parameters is obtained.In addition, all of the researches above are accompanied with computer experiments. The simulations results not only demonstrate the evolutionary tendency of the solutions, but also validate theoretical analysis results.