一类四阶有理差分方程的振动规律和全局稳定性

Rules of Oscillation and Global Asymptotic Stability for a Fourth-Order Rational Difference Equation

主要讨论了一类四阶有理差分方程xn+1=(xn-2+xn-3)/(xn-2xn-3+1),n=0,1,2,…,初始值x-3,x-2,x-1,x0∈(0,∞)的振动规律和全局稳定性,即描述了其解的振动周期为15,且正、负半环长的规律为:4+,3-,1+,2-,2+,1-,1+,1-;又指出了解之间存在xn+kΔ(C(xn+k))x(nC(xn+k)C(xn))(n≥-3)的大小关系;并得到了方程的平衡点是全局渐近稳定的.

A fourth-order rational difference equation is considered. The rules of oscillation and global asymptotic stability is described clearly. Mainly, the lengths of positive and negative semi-cycles of its nontrivial solutions are found to occur periodically with prime period 15. The rule is 4^＋ ,3^- ,1^＋ ,2^- ,2^＋ ,1^- ,1^＋ ,1^- in a period. The relation between solution and solution is xn＋kΔ （C （xn＋k ） ）xn^（C（xn＋k）C（xn）） （n ≥ -3 ）. By utilizing this rule and relation, its negative equilibrium point is verified to be globally asymptotically stable.