一类差分方程的轨道结构和全局渐近稳定性

研究了一类五阶有理差分方程xn 1=F(xn,xn-1,xn-3,xn-4)/G(xn,xn-1,xn-3,xn-4),其中F(xn,xn-1,xn-3,xn-4)=xnxn-1 xnxn-3 xnxn-4 xn-1xn-3 xn-1xn-4 xn-3xx-4 xnxn-1xn-3xn-4 1,G(xn,xn-1,xn-3,xn-4)=xn xn-1 xn-3 xn-4 xnxn-1xn-3 xnxn-1xn-4 xnxn-3xn-4 xn-1xn-3xn-4,初值x-4,x-3,x-2,x-1,x0∈(0, ∞),a∈[0,∞),N=0,1,….研究表明:随着初值的变化,该方程非平凡解的正、负半环长度规律为…,4 ,2,1 ,1-,1 ,1,2 ,1,3 ,1,1 ,5,1 ,2,2 ,3-,4 ,2,1 ,1-,1 ,1,2 ,1,3 ,1,1 ,5,1 ,2,2 ,3-….利用这个规律,证明了该方程的正平衡点是全局渐近稳定的.

The Rule of Trajectory Structure and Global Asymptotic Stability for a Kind of Nonlinear Difference Equation

The fifth-order rational difference equation is xn+1=F(xn,xn-1,xn-3,xn-4)/G(xn,xn-1,xn-3,xn-4),where F(xn,xn-1,xn-3,xn-4)=xnxn-1+xnxn-3+xnxn-4+xn-1xn-3+xn-1xn-4+xn-3xx-4+xnxn-1xn-3xn-4+1, G(xn,xn-1,xn-3,xn-4)=xn+xn-1+xn-3+xn-4+xnxn-1xn-3+xnxn-1xn-4+xnxn-3xn-4+xn-1xn-3xn-4,The initial values x-4,x-3,x-2,x-1,x0∈(0,+∞),a∈[0,∞),N=0,1,….It is found that, with change of the initial values, the rule for the lengths of positive and negative semi-cycles for nontrivial solutions of this equation to successively occur is…,4+,2,1+,1-,1+,1,2+,1,3+,1,1+,5,1+,2,2+,3-,4+,2,1+,1-,1+,1,2+,1,3+,1,1+,5,1+,2,2+,3-…By the use of the rule, we proved that the positive equilibrium point of the equation is globally asymptotically stable.