Global Dynamics Of Two Rational Difference Equations With Higher Order

The studies of difference equations,including the stability,oscillation,boundary value problems,and homoclinic and heteioclinic orbits,etc,have undergone rapid development in the past decades.Difference equations also have very strong backgrounds in practical applications,such as mathematical biological models,mathematical ecological models,engineering problems.In this thesis,we prove that the unique non-negative equilibrium points of two classes of higher order rational difference equations are globally attractive.As application,our results not only improve many known results,but also solve several "Open Problems and Conjectures".In the first chapter,we introduce the history,background,current status,current developments and conjectures for the study of rational difference equations,and introduce the origin and purpose in this thesis.In Chapter 2,some definitions of difference equation are listed,and the basic knowledge and several key lemmas are used in the proof of main results in this thesis.In Chapter 3,the global dynamical properties of a class of rational difference equa tion xn+1=p+qxn/(1+rxn-k)+(sxn-l)are discussed.The results obtained dispaly that the nonnegative equilibrium solution of the difference equation is globally attractive for any nonnegative parameters p,q,r,s,and positive number A.This result answers serval "Open Problems and Conj ectures" posed by G.Ladas et al.In Chapter 4,another rational difference equation xn+1=p+qxn/(1+rxn)+(sxn-k)is explored and the global attractivity of the equilibrium solution of the equation is obtained in two different cases.The result improves some existing ones.In the fifth Chapter,some applications of the two classes of difference equations are presented when the parameters vary,and the conclusions obtained improve some known conclusions and solve serval conjectures.In the last section,the whole thesis is summarized and some prospects for further research are presented.