Dynamics Of Several Classes Of Recursive Sequences

In this thesis the dynamics, i.e., the global stability, the period character and the bound-edness nature of the solutions, of several classes of difference equations, are systematically studied, which confirm some conjectures. We organize the thesis as follows:In chapter one, the historic background of the difference equations is introduced briefly.In chapter two, some basic definitions and results of the difference equations which will be used in context are introduced.In chapter three, we investigate the dynamics of the difference equationwhere p > 0, and the initial conditions x_-2, x_-1, x₀∈ [0, ∞) such that the denominator of the equation is never zero. We establish the following:(1) The unique nonnegative equilibrium point x|- = 1/(1+p) is globally asymptotically stable if(2) Every solution of the equation converges to a period-two (not necessarily prime) solution ifp=1;(3) The unique nonnegative equilibrium point x|- = 1/(1+p) is unstable (in fact, a saddle point) ifp> 1.In chapter four, we investigate the stability character of the difference equationwhere α,β,γ, A, B, C ∈ (0,∞) and the initial conditions x_-1, x₀ ∈ [0,∞). We establish the following:(1) Suppose that the equation has no prime period-two solution andThen the positive equilibrium of the equation is locally asymptotically stable.